Rational points and generalized trace forms on a finite algebra over a real closed field

TL;DR

This paper provides a new proof of the Pederson-Roy-Szpirglas theorem by connecting counting real solutions of polynomial equations to invariants of trace forms on finite algebras over real closed fields.

Contribution

It offers a novel proof of a counting theorem using linear algebra and algebraic invariants, linking algebraic geometry and form theory.

Findings

Proof of Pederson-Roy-Szpirglas theorem using trace form signatures

Establishes equality between rational points count and trace form signature

Connects algebraic invariants with real zero counting

Abstract

The main goal of this article is to provide a proof of the Pederson-Roy-Szpirglas theorem about counting common real zeros of real polynomial equations by using basic results from Linear algebra and Commutative algebra. The main tools are symmetric bilinear forms, Hermitian forms, trace forms, and their invariants such as rank, types, and signatures. Further, we use the equality (proved in [3]) of the number of K-rational points of a zero-dimensional affine algebraic set over a real closed field $K$ with the signature of the trace form of its coordinate ring to prove the Pederson-Roy-Szpirglas theorem, see [16].

Equations

$${\mathrm{Sesq}}_K(V,W)⟶K^{I\times J}={\mathrm{M}}_{I,J}(K),\Phi ⟼{\left(\Phi (x_i,y_j)\right)}_{(i,j)\in I\times J},\text{vspace}\ast -2mm$$

$$\displaystyle\Phi:\bigl{(}\bigoplus_{i\in I}V_{i}\bigr{)}\times\bigl{(}\bigoplus_{i\in I}W_{i}\bigr{)}\longrightarrow K\,,\,\left((x_{i}),(y_{i})\right)\longmapsto\sum_{i\in I}\Phi_{i}(x_{i},y_{i})\vspace*{-2mm}$$

$$D_0:=1and{D}_i:=\left|\begin{array}{ccc}\Phi (x_1,x_1)& \cdots & \Phi (x_1,x_i)\\ ⋮& \ddots & ⋮\\ \Phi (x_i,x_1)& \cdots & \Phi (x_i,x_i)\end{array}\right|,i=1,\dots ,n,\text{vspace}\ast -2mm$$

$${\mathcal{G}}_{{\Phi }_h}(1,x)=\left(\begin{array}{cc}h(z)+h(\overline{z})& h(z)\cdot z+h(\overline{z})\cdot \overline{z}\\ h(z)\cdot z+h(\overline{z})\cdot \overline{z}& h(z)\cdot z^2+h(\overline{z})\cdot {\overline{z}}^2\end{array}\right)\in {\mathrm{M}}_2(K)$$

$$K^n$$

$$a$$

$$\left(\begin{array}[]{cc}\operatorname{Tr}\,(1)&\operatorname{Tr}\,({\rm i})\\ \operatorname{Tr}\,({\rm i})&\operatorname{Tr}\,(-1)\end{array}\right)\,=\,\left(\begin{array}[]{cr}2&0\\ 0&-2\end{array}\right)\,.$$

Full text

† This expository article on the Pederson-Roy-Szpirglas theorem about counting real roots of real polynomial equations. Most of the exposition is influenced by the discussions of the first author with late Prof. Dr. Uwe Storch (1940-2017) and the lecture courses delivered by him.

∗ During the preparation of this work the first author was visiting Department of Mathematics, Indian Institute of Technology Bombay. He would like to express his gratitude for the generous financial support from IIT Bombay and encouraging cooperation. He was also partially supported by his project MATRICS-DSTO-1983 during the final preparation of this manuscript.

1. Introduction

The objective of this paper is to present an exposition of classical and modern results con- cerning the number of real or complex points in the solution space of a finite system of polynomial equations with real coefficients in arbitrary number of variables. More precisely, for polynomials $F_{1},\ldots,F_{m}\in\mathds{R}[X_{1},\ldots,X_{n}]$, assume that the residue-class $\mathds{R}$-algebra $\mathds{R}[X_{1},\ldots,X_{n}]/\langle F_{1},\ldots,F_{m}\rangle$ is finite dimensional over $\mathds{R}$, then the set of common zeros

       $\,{\rm V}_{\mathds{R}}(F_{1},\ldots,F_{m}):=\{(a_{1},\ldots,a_{n})\in\mathds{R}^{n}\mid F_{j}(a_{1},\ldots,a_{n})=0\ \hbox{ for all }\ j=1,\ldots,m\}\,$

of $F_{1},\ldots,F_{m}$ in $\mathds{R}^{n}\!$ is finite. The converse is not true, for example, for $F_{1}\!=\!X_{1}^{2}\!+\!1$, ${\rm V}_{\mathds{R}}(F_{1})\!=\!\emptyset$ is finite and \mathds{R}[X_{1},\ldots,X_{n}]/\langle F_{1}\rangle\!\!\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}\!\!\mathds{C}[X_{2},\ldots,X_{n}] is not finite dimensional over $\mathds{R}$ if $n\!\geq\!2$. However, for polynomials $F_{1},\ldots,F_{m}\!\in\!\mathds{C}[X_{1},\ldots,X_{n}]$, the residue-class $\mathds{C}$-algebra $\mathds{C}[X_{1},\ldots,X_{n}]/\langle F_{1},\ldots,F_{m}\rangle$ is finite dimensional over $\mathds{C}$ if and only if the set of common zeros ${\rm V}_{\mathds{C}}(F_{1},\ldots,F_{m})$ of $F_{1},\ldots,F_{m}$ in $\mathds{C}^{n}$ is finite. Moreover, by the classical Hilbert’s nullstellensatz ${\rm V}_{\mathds{C}}(F_{1},\ldots,F_{m})\neq\emptyset$ if and only if the ideal $\langle F_{1},\ldots,F_{m}\rangle$ generated by $F_{1},\ldots,F_{m}$ in $\mathds{C}[X_{1},\ldots,X_{n}]$ is a non-unit ideal. But, this is not true over the field $\mathds{R}$ or more generally over real closed fields. Therefore the natural questions one deals with are when exactly ${\rm V}_{K}(F_{1},\ldots,F_{m})\neq\emptyset$ and how to find its cardinality, where $K$ is an arbitrary real closed field.

Many researchers have studied these problems and devised effective algorithms. For example, already in the 19th century Sturm, Jacobi, Sylvester, Hermite, Hurwitz proved fundamental results for counting real points (in small number of variables $n\!\leq\!2$) by using the signature of appropriate quadratic forms.

In Section 2 and Section 3, we collect standard results on symmetric bilinear and Hermitian forms over a real closed field $K$ and its algebraic closure $\mathds{C}_{K}\!=\!K[\,{\rm i}\,]$ with ${\rm i}^{2}\!=\!-1$. However, for the sake of completeness, we recall them without proofs in the format they are used in later sections. With these preliminaries at the end od Section 3, we state the important Rigidity theorem for quadratic forms (see [3]) which is used in Section 4.

In Section 4, we collect some elementary concepts from commutative algebra and recall the important Theorem 4.5 from [3] which relates the $K$-rational points of a finite dimensional algebra $A$ over a real closed field $K$ with the type of the trace form $\operatorname{Tr}_{K}^{A}$ on $A$ and derive some consequences.

In Section 5, we compute the cardinality of the $K$-rational points of finite algebra over real closed field $K$. The main ingredient in this section is the Shape Lemma 5.3 which guarantees a distinguished generating set for a [math]-dimensional radical ideal $\mathfrak{A}\subseteq K[X_{1},\ldots,X_{n}]$ from which one can reduce the problem of counting the number of $K$-rational points in ${\rm V}_{K}(\mathfrak{A})$ to the one variable case. In Theorem 5.5 using the results from Section 4, we relate type, signature and rank of a generalized trace forms on $A=K[X_{1},\ldots,X_{n}]/\mathfrak{A}$ with the number of points in ${\rm V}_{K}(\mathfrak{A})$ and in ${\rm V}_{\overline{K}}(\mathfrak{A})$. Finally, we give a proof of theorem of Pederson-Roy-Szpirglas [16, Theorem 2.1].

2. Decomposition theorem for Hermitian forms

The main aim of this section is to recall the Decomposition Theorem (see 2.13) which guarantees the existence of orthogonal bases (with respect to Hermitian forms). For this we recall basic concepts and steps which lead to its proof. Most of these results can be found in standard graduate text books, for instance see [18, Ch. V, §12], [17, Ch. IX] or [9, Ch. 11], [1, Ch. 7], or [14, Ch. XV]. However, for setting the notation, terminology and for the sake of completeness, we recall them without proofs in the format that they are used in this as well as in the later sections.

2.1 Notation and Assumptions

In order to define symmetric and Hermitian forms together and prove results about them, we fix the following convenient notation :

*Let $K$ be a field and let $\kappa:K\rightarrow K$ be a fixed involution (an automorphism whose square is the identity, i .e. its inverse is itself). We denote by $K^{\prime}:=K^{\kappa}:=\{a\in K\mid\kappa(a)=a\}\subseteq K$ the fixed field of $K$. There are exactly two cases : (i)  $\kappa=\operatorname{id}_{K}$ and (ii)  $\kappa\neq\operatorname{id}_{K}$. In this first case, we assume that ${\rm Char}\,K\neq 2$.

The involution $\kappa$ of $K$ is simply denoted by the standard bar-notation $\kappa:K\rightarrow K$, $a\mapsto\overline{a}$ and called the conjugation of $K$. Therefore we have : $\,\overline{a+b}=\overline{a}+\overline{b}$, $\,\overline{ab}=\overline{a}\overline{b}$ and $\overline{\overline{a}}=a$ for all $a,b\in K$. Furthermore, the fixed field $K^{\prime}=K$ in the first case and $K^{\prime}\subsetneq K$ in the second case.*

2.2 Examples

• (1)

For an arbitrary field $K$, the identity map $\operatorname{id}_{K}:K\to K$ is an involution. For $K\!=\!\mathds{R}$, the identity $\operatorname{id}_{\mathds{R}}\!$ is the only involution. For $K\!=\!\mathds{C}$, besides the identity $\operatorname{id}_{\mathds{C}}$, the usual complex conjugation $\mathds{C}\rightarrow\mathds{C}$, $z\mapsto\overline{z}$, is the only other involution of $\mathds{C}$ which play an important role 111 In the case, $V=W=\mathds{C}$, the distance of a point $z$ from the origin is not given by the bilinear form $(z,w)\mapsto z\cdot w$, but by using the map $(z,w)\mapsto z\cdot\overline{w}\,$, namely, $|z|=\sqrt{z\,\overline{z}}$..

• (2)

*The complex-conjugation is a special case of the conjugation of a quadratic algebra $A$ over an arbitrary field $K$ : If $1,\omega\in A$ is a $K$-basis of $A$ with $\omega^{2}=\alpha+\beta\omega$, $\alpha$, $\beta\in K$, then the conjugation $A\rightarrow A$ of $A$ is defined by $\overline{a+b\,\omega}=(a+b\beta)-b\,\omega$, $a$, $b\in K$. It is easy to see this is an involution of the $K$-algebra $A$ and is not equal to $\operatorname{id}_{A}$. For an arbitrary element $x\in A$, the norm, the trace and the characteristic polynomial of $x$ over $K$ are defined by the equations :  ${\rm N}^{A}_{K}(x)=x\overline{x}$, $\,\operatorname{Tr}_{K}^{A}(x)=x+\overline{x}$, $\,\chi_{x}=X^{2}-(x+\overline{x})\,X+x\overline{x}=(X-x)(X-\overline{x})$, respectively. *

There are many examples of this type, for example, if $L$ is a field and if $\kappa\in\textrm{Aut}\,L$ is an involution of $L$ with $\kappa\neq\operatorname{id}_{L}$, then $L$ is a quadratic algebra over the fixed field $K:=L^{\kappa}:=\{a\in L\mid\kappa(a)=a\}$ and the involution $\kappa$ of $L$ coincides with the conjugation of the quadratic algebra over $K$ and the Galois group $\,\operatorname{Gal}(L\,|\,L^{\kappa})=\textrm{Aut}_{L^{\kappa}\hbox{\scriptsize-alg}}\,L=\{\operatorname{id}_{L},\kappa\}$. A typical example of this type is the algebraic closure $\mathds{C}_{K}=K[\,{\rm i}\,]$, where ${\rm i}^{2}=-1$, of a real closed field $K$.*

2.3 Definitions

Let $V$ and $W$ be $K$-vector spaces.

• (1)

A map $f:V\rightarrow W$ of $K$-vector spaces is called semilinear (or conjugate-linear) (with respect to the conjugation of $K$) if $f$ is additive and $f(ax)=\overline{a}x\,$ for all $a\in K$ and all $x\in V$. The semilinear maps from $V$ into $W$ coincide with the $K$-linear maps from $V$ into the anti-vector space $\overline{W}$ corresponding to $W$ and also with the $K$-linear maps from the $K$-vector space $\overline{V}$ into $W$, where $\overline{W}$ (resp. $\overline{V}$) is a $K$-vector spaces with the scalar multiplication $(a,y)\mapsto\overline{a}y$ defined by using the given scalar multiplication on $W$ (resp. $V$).

• (2)

A function $\Phi\colon V\times W\to K$ is called sesquilinear if $\Phi$ is $K$-linear in the first component and semilinear (with respect to the conjugation of $K$) in the second component, i. e. if for all $a$, $a^{\prime}\in K$ and all $x$, $x^{\prime}\in V$, $y$, $y^{\prime}\in W$, we have :

• (a)

$\,\Phi(ax+a^{\prime}x^{\prime},y)=a\,\Phi(x,y)+a^{\prime}\,\Phi(x^{\prime},y)$.

• (b)

$\,\Phi(x,ay+a^{\prime}y^{\prime})=\overline{a}\,\Phi(x,y)+\overline{a^{\prime}}\,\Phi(x,y^{\prime})$.

The set of sesquilinear functions $V\times W\to K$ is denoted by ${\rm Sesq}_{K}(V,W)$ which is clearly a subspace of the $K$-vector space $K^{V\times W}\!$. If $V=W$, then sesquilinear functions are also called sesquilinear forms on $V$. Note that if the conjugation of $K$ is equal to $\operatorname{id}_{K}$, then the sesquilinear functions are linear in both variables, i. e. they are bilinear and hence $\operatorname{Sesq}_{\,K}(V,W)=\operatorname{Mult}_{K}(V,W)$ ( = the set of bilinear functions).

The bijective map $\operatorname{Sesq}_{\,K}(V,W)\longrightarrow(V\otimes_{K}\overline{W})^{\ast}$, $\,\Phi\longmapsto\left(x\otimes y\mapsto\Phi(x,y)\right)$ is an isomorphism of $K$-vector spaces (with inverse $\,\varphi\longmapsto\left((x,y)\mapsto\varphi(x\otimes y)\right)$.

2.4

Gram’s Matrix*  Let $V$, $W$ be finite dimensional $K$-vector spaces with bases $\mathbcal{x}:=\{x_{i}\mid i\in I\}$, $I$ finite indexed set, $\mathbcal{y}:=\{y_{j}\mid j\in J\}$, $J$ finite indexed set, respectively. Then every sesquilinear function $\Phi\colon V\times W\rightarrow K$ is uniquely determined by the values $\Phi(x_{i},y_{j})$, $(i,j)\in I\times J$. Conversely, for arbitrary family $c_{ij}\in K$, $(i,j)\in I\times J$, there is a (unique) sesquilinear function $\Phi\colon V\times W\to K$ defined by $\Phi(\sum_{i\in I}a_{i}x_{i},\sum_{j\in J}b_{j}y_{j}):=\sum_{(i,j)\in I\times J}a_{i}\,{b}_{j}\,c_{ij}$. Moreover, the map*

[TABLE]

*is an isomorphism of $K$-vector spaces.

For a sesquilinear function $\Phi:V\times W\longrightarrow K$ the $I\times J$-matrix

                           $\displaystyle\,\mathscr{G}_{\Phi}(\mathbcal{x}\,;\,\mathbcal{y}):=\left(\Phi(x_{i},y_{j})\right)_{(i,j)\in I\times J}\in{\rm M}_{I,J}(K)\,$

is called the Gram’s matrix or the fundamental matrix of $\Phi$ with respect to the bases $\mathbcal{x}$ and $\mathbcal{y}$. If $V=W$ and $\mathbcal{x}=\mathbcal{y}$, then we simply write $\,\mathscr{G}_{\Phi}(\mathbcal{x})$. Further, if $I=J$, then the determinant ${\rm G}_{\Phi}(\mathbcal{x},\mathbcal{y}):=\operatorname{Det}\,\mathscr{G}_{\Phi}(\mathbcal{x}\,;\,\mathbcal{y})$ is called the Gram’s determinant with respect to the bases $\mathbcal{x}$ and $\mathbcal{y}$. If $V=W$ and if $y_{j}=x_{j}$ for all $j\in J=I$, we simply write ${\rm G}_{\Phi}(\mathbcal{x})$.*

*For the computation with Gram’s matrices, it is convenient to extend the conjugation of $K$ to matrices over $K$ :  For a matrix $\mathscr{A}=\left(a_{ij}\right)\!\in\!{\rm M}_{I,J}(K)$ with $I$, $J$ finite indexed sets, put $\,\overline{\mathscr{A}}\!:=\!\left(\overline{a}_{ij}\right)\!\in\!{\rm M}_{I,J}(K)$. Then the map ${\rm M}_{I,J}(K)\rightarrow\!{\rm M}_{I,J}(K)$, $\mathscr{A}\mapsto\overline{\mathscr{A}}$, is a semilinear (with respect to the conjugation of $K$) involution of the $K$-vector space ${\rm M}_{I,J}(K)$. Further, ${}^{\rm t}\overline{\mathscr{A}}=\overline{{}^{{\rm t}}{\mathscr{A}}}$ (where for a $I\times J$-matrix $\mathscr{A}\in\operatorname{M}_{I,J}(K)$, ${}^{\rm t}\mathscr{A}$ denote the transpose of $\mathscr{A}$) and $\overline{\mathscr{A}\mathscr{B}}=\overline{\mathscr{A}}\,\overline{\mathscr{B}}$ if $\mathscr{B}\in{\rm M}_{J\,R}(K)$, $R$ finite indexed set. For a square matrix $\mathscr{A}\in{\rm M}_{I}(K)$, $\operatorname{Det}\overline{\mathscr{A}}=\overline{\operatorname{Det}\mathscr{A}}$ and if $\,\mathscr{A}\in\operatorname{GL}_{I}(K)$, then $\overline{\mathscr{A}}^{-1}\!=\!\overline{\mathscr{A}^{-1}}$.

2.4.1  Let $\mathbcal{x}^{\prime}=(x^{\prime}_{i})_{i\in I}$, $\mathbcal{y}^{\prime}=(y^{\prime}_{j})_{j\in J}$ be another $K$-bases of $V$, $W$ and $\mathscr{A}=(a_{ri})\in\operatorname{GL}_{I}(K)$, $\mathscr{B}=(b_{sj})\in\operatorname{GL}_{J}(K)$ be the transition matrices of the bases from $\mathbcal{x}$ to $\mathbcal{x}^{\prime}$, from $\mathbcal{y}$ to $\mathbcal{y}^{\prime}$, respectively. Then for a sesquilinear function $\Phi:V\times W\rightarrow K$, we have the transformation formula $:$ 

                $\,{\mathscr{G}}_{\Phi}({\mathbcal{x}}\,;\,{\mathbcal{y}})={}^{{\rm t}}{\mathscr{A}}\,{\mathscr{G}}_{\Phi}({\mathbcal{x}}^{\prime}\,;\,{\mathbcal{y}}^{\prime})\,\overline{\mathscr{B}}\quad\hbox{or}\quad{\mathscr{G}}_{\Phi}({\mathbcal{x}}^{\prime}\,;\,{\mathbcal{y}}^{\prime})={}^{{\rm t}}{\mathscr{A}}^{-1}\,{\mathscr{G}}_{\Phi}({\mathbcal{x}}\,;\,{\mathbcal{y}})\,\overline{\mathscr{B}}^{-1}\!\!.$

In particular, if $V=W$, then $\,{\mathscr{G}}_{\Phi}(\mathbcal{x})\!={}^{\rm t}{\mathscr{A}}\,{\mathscr{G}}_{\Phi}({\mathbcal{x}}^{\prime})\,\overline{\mathscr{A}}$.*

2.5 Examples

• (1)

(Standard forms*)  The standard form on the standard $K$-vector space $K^{(I)}$, $I$ an indexed set, with the standard basis $e_{i},i\in I$ of $K^{(I)}$ is the sesquiliner form defined by the unit matrix $\mathscr{E}_{I}\in{\rm M}_{I}(K)$ and is denoted by $\langle-,-\rangle$, that is, $\langle e_{i},e_{j}\rangle=\delta_{ij}$ for all $i,j\in I$. Therefore \,\bigr{\langle}(a_{i})\,,(b_{i})\,\bigr{\rangle}=\sum_{i\in I}a_{i}\,\overline{b}_{i}={}^{\rm t}{\mathscr{A}}\,\mathscr{B}\,, where $\,{\mathscr{A}}:=(a_{i})\,,{\mathscr{B}}:=(b_{i})\in K^{(I)}$ (are column vectors). In particular, if $I=\{1,\ldots,n\}$, then $\langle(a_{1},\ldots,a_{n}),(b_{1},\ldots,b_{n})\rangle=a_{1}\overline{b}_{1}+\cdots+a_{n}\overline{b}_{n}$.*

• (2)

(Natural Duality)*  Let $K$ be an arbitrary field with $\operatorname{id}_{K}$ as conjugation, $V$ a $K$-vector space and let $V^{*}={\rm Hom}_{K}\,(V,K)$ denote the dual space of $V$. The canonical evaluation map $\,\mathscr{E}:V\times V^{*}\longrightarrow K$, $\,(x,f)\longmapsto\langle x,f\rangle:=f(x)$, is a bilinear and is called the natural duality between $V$ and $V^{*}$. If $V$ is finite dimensional with basis $\mathbcal{x}=\{x_{1},\ldots,x_{n}\}$, and if ${\mathbcal{x}}^{*}=\{x_{1}^{*},\ldots,x^{*}_{n}\}$ is the corresponding dual basis, then the Gram’s matrix of this natural duality $\mathscr{G}_{\mathscr{E}}(\mathbcal{x}\,,\,{\mathbcal{x}}^{*})=\mathscr{E}_{n}$ is the unit matrix in $\operatorname{M}_{n}(K)$.*

2.6

Non-degeneracy and Complete Duality*  An important motivation for the study of sesquilinear functions is the description of linear form through vectors. (See Example 2.9).*

*Let $V$ and $W$ be $K$-vector spaces and let $\Phi\colon V\times W\longrightarrow K$ be a sesquilinear function. The canonical semilinear maps defined by :  

    $\,\Phi_{1}\colon V\longrightarrow W^{*}\,\ x\longmapsto(\,y\mapsto\overline{\Phi(x,y)}\,)\quad{\rm and}\quad\Phi_{2}\colon W\longrightarrow V^{*}\,\ y\longmapsto\left(\,x\mapsto\Phi(x,y)\,\right)$,

are simple denoted by $\Phi_{1}(x)=\Phi(x,-)$ and $\Phi_{2}(y)=\Phi(-,y)$.*

Further, from each one of the map $\Phi_{1}$ resp. $\Phi_{2}$, one can recover $\Phi$, since $\,\Phi(x,y)=\overline{\left(\Phi_{1}(x)\right)(y)}=\left(\Phi_{2}(y)\right)(y)\,$ for all $x\in V$ and all $y\in W$.

2.6*.1  Suppose that both $V$, $W$ are finite dimensional over $K$ with bases ${\mathbcal{x}}=(x_{i})_{i\in I}$, ${\mathbcal{y}}=(y_{j})_{j\in J}$, respectively. Then the matrices of the canonical semilinear maps $\Phi_{1}$ and $\Phi_{2}$ with respect to bases $\mathbcal{x}$, ${\mathbcal{y}}^{*}$ and $\mathbcal{y}$, ${\mathbcal{x}}^{*}$, where ${\mathbcal{x}}^{*}$ and ${\mathbcal{y}}^{*}$ are dual bases of $\mathbcal{x}$ and $\mathbcal{y}$, respectively, are given by $:$

                           $\,{\mathscr{M}}^{{\mathbcal{x}}}_{{\mathbcal{y}}^{*}}(\Phi_{1})={}^{\rm t}\overline{{\mathscr{G}}_{\Phi}({\mathbcal{x}}\,;\,{\mathbcal{y}})}\quad\hbox{and}\quad{\mathscr{M}}^{{\mathbcal{y}}}_{{\mathbcal{x}}^{*}}(\Phi_{2})={\mathscr{G}}_{\Phi}({\mathbcal{x}}\,;\,{\mathbcal{y}})$.

Further, since taking the transpose and conjugation, the rank of a matrix is unaltered, both $\Phi_{1}$ and $\Phi_{2}$ have the same rank. This common rank of the maps $\Phi_{1}$, $\Phi_{2}$ is called the rank of the sesquilinear function $\Phi$ and is denoted by ${\rm rank}\,\Phi$. Therefore, $\,{\rm rank}\,\Phi$ is the rank of the Gram’s matrix of $\Phi$ with respect to arbitrary bases of $V$ and $W$.*

The case when $\Phi_{1}$ and $\Phi_{2}$ are both injective or both bijective are important :

2.7 Definition

*Let $V$ and $W$ be $K$-vector spaces and let $\Phi\colon V\times W\longrightarrow K$ be a sesquilinear function. We say that

(1) $\,\Phi$ is non-degenerate if $\Phi_{1}$ and $\Phi_{2}$ are both injective.

(2) $\,\Phi$ defines a complete duality (between $V$ and $W$) if $\Phi_{1}$ and $\Phi_{2}$ are both bijective.*

2.8 Example

(Trace form)*  Let $V$ be a finite dimensional $K$-vector space. The map $\operatorname{End}_{K}V\times\operatorname{End}_{K}V$, $\,(f,g)\longmapsto\operatorname{Tr}(fg),$ is a symmetric bilinear form on the $K$-vector space ${\rm End}_{K}V$ of $K$-endomorphisms of $V$ and is called the trace form on ${\rm End}_{K}V$.*

2.8* 1  Let $V$ be a finite dimensional $K$-vector space. Then the trace form defines a complete duality on ${\rm End}_{K}V$.*

Let $A$ be a finite (dimensional) $K$-algebra. For an element $x\in A$, $\lambda_{x}:A\to A$ denote the left multiplication map on $A$ by $x$. Then the map $A\times A\to K$, $\,(x,y)\mapsto\operatorname{Tr}_{K}^{A}(xy)=\operatorname{Tr}(\lambda_{x}\lambda_{y})$, defines a symmetric bilinear form on $A$ and is called the trace form of the $K$-algebra $A$. The trace form on $A$ reflects many important properties of the $K$-algebra $A$, see Section 4.

Note that if $A={\rm End}_{K}V$ , then the trace form on the $K$-algebra ${\rm End}_{K}V$ is different from the above introduced trace form on the $K$-vector space ${\rm End}_{K}V$. Obviously, for every endomorphism $f\in{\rm End}_{K}V$ : $\,\operatorname{Tr}_{K}^{{\rm End}_{K}V}f=n\,\cdot\operatorname{Tr}f\,$, $\enskip n:={\rm Dim}_{K}V\,$.

2.9 Example

*( Gradient of a linear form )  For a finite dimensional $K$-vector space $V$ the natural duality between $V$ and $V^{*}$ (see Example 2.5 (2)) is a complete duality, since its Gram’s matrix with respect to dual bases is the unit matrix. Further, in this case $\Phi_{1}:V\rightarrow(V^{*})^{*}$ is the canonical evaluation map $x\mapsto\mathscr{E}_{x}:f\mapsto\mathscr{E}_{x}(f):=f(x)$ and $\Phi_{2}:V^{*}\rightarrow V^{*}$ is the identity map $\operatorname{id}_{V^{*}}$. In particular, for every linear form $\varphi\colon V^{*}\to K$, there exists a unique vector $x\in V$ (which is independent on $\varphi$) such that $\varphi(f)=f(x)$ for every $f\in V^{*}$, i. e. $\varphi$ is the evaluation of the linear forms in $V^{*}$ at the vector $x\in V$.

If $V$ is not finite dimensional, then the natural duality is non-degenerate but never complete. This follows from the fact that for every $x\in V$ (also in the infinite dimensional case) can be extended to a basis $V$ and hence there exists a linear form $f\in V^{*}$ with $\langle x,f\rangle=f(x)\neq 0$.*

More generally, if $\,\Phi:V\times W\longrightarrow K$ defines a complete duality, then one can use $\Phi_{1}$ and $\Phi_{2}$ to identify the $K$-vector spaces $V$ and $W^{*}\!\!$, resp. $W$ and $V^{*}\!\!$. Therefore, for every linear form $f\in W^{*}\!\!$, there exists a unique vector $x_{f}\in V$ with $f=\overline{\Phi(x_{f},-)}$, and for every linear form $\varphi\in V^{*}\!\!$, there exists a unique vector $y_{\varphi}\in W$ with $\varphi=\Phi(-,y_{\varphi})$. The vectors $x_{f}$ resp. $y_{\varphi}$ are called the gradients of $f$ resp. $\varphi$ (with respect to $\Phi$) and are denoted by ${\rm grad}\,f$ resp. ${\rm grad}\,\varphi$. Therefore the linear forms on $W$ resp. $V$ correspond to their respective gradients, i. e. $f=\Phi_{1}({\rm grad}\,f)$ and $\varphi=\Phi_{2}({\rm grad}\,\varphi)$.

2.10

Orthogonality, perpendicular relation and Hermitian forms*  The concept of orthogonality has its origin in Euclidean geometry.*

*Let $V$ and $W$ be $K$-vector spaces and let $\Phi\colon V\times W\longrightarrow K$ be a sesquilinear function.

(1)  The vectors $x\in V$ and $y\in W$ are called orthogonal or perpendicular to each other with respect to $\Phi$ if $\Phi(x,y)=0$. In this case we write $x\bot_{\Phi}\,y$ or simply $x\bot y$ (if $\Phi$ is fixed).

(2) Two subsets $M\subseteq V$ and $N\subseteq W$ are called orthogonal if $x\bot y$ for all $x\in M$ and for all $y\in N$. In this case we write $M\bot N$. Futher, we put

                       $M^{\bot}:=\{y\in W\mid M\bot\{y\}\}\quad\hbox{and}\quad{}^{\bot}N:=\{x\in V\mid\{x\}\bot N\}$.

Obviously, $M^{\bot}$ and ${}^{\bot}\,N$ are $K$-subspaces of $W$ and $V$, resp.*

For example, if $f\in W^{*}$ is a linear form with a gradient (see Example 2.8) ${\rm grad}\,f\in V$, i. e. $f(y)=\overline{\Phi({\rm grad}\,f,y)}$ for all $y\in W$, then $\operatorname{Ker}f=\{{\rm grad}\,f\}^{\bot}$. Analogously, if a linear form $\varphi\in V^{*}$ with a gradient ${\rm grad}\,\varphi\in W$, then $\operatorname{Ker}\varphi={}^{\bot}\,\{{\rm grad}\,\varphi\}$.

Note that if $\Phi:V\times V\longrightarrow K$ is a sesquilinear form on $V$, then for a subset $M\subseteq V$, the subsets $M^{\bot}\!=\!\{y\in V\mid M\bot\{y\}\}$ and ${}^{\bot}\,M\!=\!\{x\in V\mid\{x\}\bot M\}$ are not equal in general, since the relation of perpendicularity is not symmetric. To remove this difference, one considers the symmetric (resp. Hermitian, skew-Hermitian) forms if the conjugation of $K$ (see 2.1) is $\operatorname{id}_{K}$ (resp. $\neq\operatorname{id}_{K}$) :

2.10*.1  Definition  Let $V$ be a finite dimensional $K$-vector space and let $\Phi:V\times V\longrightarrow K$ be a sesquilinear form on $V$, We say that $\Phi$ is Hermitian (resp. skew-Hermitian) if $\Phi(x,y)=\overline{\Phi(y,x)}$ (resp. $\Phi(x,y)=-\overline{\Phi(y,x)}$) for all $x$, $y\in V$.

With the notation and assumptions in 2.1, we note that the term “Hermitian ” and “skew-Hermitian” mean “symmetric” and “skew-symmetric” if the conjugation of $K$ is $\operatorname{id}_{K}$. If the conjugation of $K$ is $\neq\operatorname{id}_{K}$, then sometimes we use the terms “pure-Hermitian” and “pure-skew-Hermitian” forms on $V$. In the case $K=\mathds{C}$ with the usual complex-conjugation, the Hermitian (resp. skew-Hermitian) forme are simply called complex-Hermitian (resp. complex-skew-Hermitian).*

2.10*.2   If $\,\Phi_{1}\colon V\longrightarrow V^{*}$ and $\Phi_{2}\colon V\longrightarrow V^{*}$ are the canonical semilinear maps associated to the sesquilinear form $\Phi$ on $V$, see 2.6, then $\Phi$ is Hermitian $($resp. skew-Hermitian$)$ if and only if $\Phi_{1}=\Phi_{2}($resp. $\Phi_{1}=-\Phi_{2})$.

Further, for a Hermitian (resp. skew-Hermitian) form $\Phi$ on $V$, the relation $\bot$ on $V$ is symmetric. In this case the subspace ${}^{\bot}V=V^{\bot}=\operatorname{Ker}\Phi_{1}=\operatorname{Ker}\Phi_{2}$ of $V$ is called the degeneration space  or,   the radical of $\Phi$ and is also denoted by ${\rm Rad}\,(V,\Phi)={\rm Rad}\,\Phi$.*

2.10*.3   A sesquilinear form $\Phi:V\times V\to K$ on a finite dimensional $K$-vector space $V$ is Hermitian $($resp. skew-Hermitian$)$ if and only if the Gram’s matrix $\mathscr{G}_{\Phi}(\mathbcal{x})=(\Phi(x_{i},x_{j}))\in{\rm M}_{I}(K)$ of $\,\Phi$ with respect to every basis $\mathbcal{x}=\{x_{i}\mid i\in I\}$ of $V$ is Hermitian $($resp. skew-Hermitian$)$.

Recall that a square matrix $\mathscr{A}\in\operatorname{M}_{I}(K)$, $I$ finite indexed set, is Hermitian (resp. skew-Hermitian if $\,\mathscr{A}={}^{\rm t}\overline{\mathscr{A}}$ (resp. $\mathscr{A}=-{}^{\rm t}\overline{\mathscr{A}}$. A matrix $\mathscr{A}\in\operatorname{M}_{I}(K)$ is symmetric (resp. skew-symmetric if $\,\mathscr{A}={}^{\rm t}{\mathscr{A}}$ (resp. $\mathscr{A}=-{}^{\rm t}{\mathscr{A}})$.*

2.10*.4   Let $V$ be a finite dimensional $K$-vector space and let ${\mathbcal{x}}\!=\!\{x_{i}\mid i\in I\}$ be a basis of $V$. (recall that $K^{\prime}$ is the fixed field of $K$, see 2.1) The map $\,\Phi\longmapsto{\mathscr{G}}_{\Phi}(\mathbcal{x})\,$ is a $K^{\prime}$-linear isomorphism of the $K^{\prime}$-vector space of Hermitian $($resp. skew-Hermitian$)$ forms on $V$ onto the $K^{\prime}$-vector space of Hermitian $($resp. skew-Hermitian$)$ matrices in ${\rm M}_{I}(K)$. Moreover, if ${\mathbcal{x}}^{\prime}=\{x^{\prime}_{i}\mid i\in I\}$ is another basis of $V$ with transition matrix $\mathscr{A}=(a_{ij})\in\operatorname{GL}_{I}(K)$, i. e.$x_{j}=\sum_{i\in I}\,a_{ij}x^{\prime}_{i}$, then the Gram’s matrices ${\mathscr{G}}_{\Phi}({\mathbcal{x}})$ and ${\mathscr{G}}_{\Phi}({\mathbcal{x}}^{\prime})$ are related by the rule :

                  ${\mathscr{G}}_{\Phi}({\mathbcal{x}})={}^{\rm t}\mathscr{A}\,{\mathscr{G}}_{\Phi}({\mathbcal{x}}^{\prime})\,\overline{\mathscr{A}}\quad$ or $\quad{\mathscr{G}_{\Phi}({\mathbcal{x}}^{\prime}})={}^{\rm t}\mathscr{A}^{-1}\,{\mathscr{G}}_{\Phi}({\mathbcal{x}})\,\overline{\mathscr{A}}^{-1}$.*

2.10*.5   In important cases a sesquilinear forms on a $K$-vector space $V$ are completely determined by its values on the diagonal \,\Delta_{\,V}=\{(x,x)\bigm{|}x\in V\}. More precisely, we have : 

2.10.5a   Polarisation identity  Let $V$ be a $K$-vector space. Then : 

(1) If the conjugation (see 2.1) of $K$ is $\neq\operatorname{id}_{K}$, then for every sesquilinear form $\Phi:V\times V\to K$, for all $x,y\in V$ and $a\in K$ with $\overline{a}\neq a$, (using Cramer’s rule) we have :

   \Phi(x,y)\!=\!\textstyle{\frac{1}{a-\overline{a}}}\,\bigl{(}\Phi(ax+y\,,ax+y)\!-\!\overline{a}\,\Phi(x+y,x+y)\!-\!\overline{a}(a\!-\!1)\,\Phi(x,x)\!-\!(1\!-\!\overline{a})\,\Phi(y,y)\bigr{)} and

   \Phi(y,x)\!=\!\textstyle{\frac{1}{\overline{a}-a}}\,\bigl{(}\Phi(ax+y\,,ax+y)\!-\!a\,\Phi(x+y,x+y)\!-\!a\,(\overline{a}\!-\!1)\,\Phi(x,x)\!-\!(1\!-\!a)\,\Phi(y,y)\bigr{)}\,.

(2) If ${\rm Char}\,K\neq 2$, then for every symmetric bilinear $\Phi:V\times V\rightarrow K$ form and for all $x,y\in V\!$, we have :  

   \Phi(x,y)\!=\!\textstyle{\frac{1}{2}}\,\bigl{(}\Phi(x+y\,,x+y)\!-\!\Phi(x,x)\!-\!\Phi(y,y)\bigr{)}\!=\!\textstyle{\frac{1}{4}}\,\bigl{(}\Phi(x+y\,,x+y)\!-\!\Phi(x-y,x-y)\bigr{)}.*

2.10*.5b   Corollary  If the conjugation of $K$ is $\neq\operatorname{id}_{K}$ (see 2.1) and $V$ is a $K$-vector space, then a sesquilinear form $\Phi:V\times V\to K$ on a $K$-vector space $V$ is Hermitian $($resp. skew-Hermitian$)$ if and only if $\Phi(x,x)=\overline{\Phi(x,x)}\,($resp. $\Phi(x,x)=-\overline{\Phi(x,x)})$ for all $x\in V$, i. e. $\Phi(x,x)\in K^{\prime}$ (the fixed field of $K$ with respect to the conjugation of $K$, see 2.1). In particular, a complex-sesquilinear form is complex-Hermitian $($resp. complex-skew-Hermitian$)$ if and only if the values the values $\Phi(x,x)$, $x\in V$, are all real $($resp. purely-imaginary$)$.*

2.10*.5c   Corollary  Let $V$ be vector space over the field $K$ of characteristic $\neq 2$. Then a symmetric bilinear form $\Phi\colon V\times V\to K$ on $V$ is the zero form if and only if $\,\Phi(x,x)=0\,\,$ for all $x\in V$.*

2.11

Orthogonal direct sums*  Let $V_{i}$, $i\in I$; $W_{i}$, $i\in I$ be two families of $K$-vector spaces and let $\,\Phi_{i}:V_{i}\times W_{i}\to K$ be a family of sesquilinear functions. Then the map*

[TABLE]

is a sesquiinear function and its restrictions $\,\Phi\,|\,V_{i}\times W_{i}=\Phi_{i}$ for all $i\in I$, where $V_{i}$ (resp. $W_{i}$) is considered canonically as subspace of $\bigoplus_{i\in I}V_{i}$ (resp. $\bigoplus_{i\in I}W_{i}$). Further, $V_{i}\bot W_{j}$ with respect to $\Phi$ for all $i,j\in I$, $i\neq j$. This sesquilinear function $\Phi$ is called the orthogonal direct sum of the family $\Phi_{i}$, $i\in I$ and is denoted by \,\displaystyle\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}_{i\in I}\Phi_{i}. Conversely, if $\Phi:V\times W\to K$ is a sesquilinear function and $V$(resp. $W$) is a direct sum of the $K$-subspaces $V_{i}$, $i\in I$ (resp. $W_{i}$, $i\in I$) with $V_{i}\perp W_{j}$ for all $i$, $j\in I$, $i\neq j$ with respect to $\Phi$, such that $\Phi(\sum_{i\in I}v_{i},\sum_{j\in I}w_{j})=\sum_{i\in I}\Phi(v_{i},w_{i})$ for $v_{i}\in V_{i}$, $w_{j}\in W_{j}$. Then we say that $\Phi$ is the orthogonal direct sum of the $\Phi_{i}:=\Phi\,|\,V_{i}\times W_{i}$, $i\in I$. In particular, if $V=W$ and $V_{i}=W_{i}$, $i\in I$, then $V$ is the orthogonal direct sum of the subspaces $V_{i}$, $i\in I$, with respect to $\Phi$ and is denoted by V=\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}_{i\in I}V_{i}.

2.12

Orthogonal basis*   Let $V$ be a $K$-vector space and let $\Phi:V\times V\rightarrow K$ be a sesquilinear form on $V$. A a family of vectors $x_{i}$, $i\!\in\!I$, in $V$ is called orthogonal with respect to $\Phi$ if $x_{i}\perp x_{j}$ for all $i$, $j\!\in\!I$ with $i\neq j$. Moreover, if $\Phi(x_{i},x_{i})\!=\!1$ for all $i\!\in\!I$, then it is called orthonormal with respect to $\Phi$.*

If $x_{i}$, $i\in I$, is an orthogonal basis of $V$ with respect to $\Phi$, then $V$ is the orthogonal direct sum of the $1$-dimensional subspaces $Kx_{i}$, $i\in I$. Moreover, if $I$ is finite, then the Gram’s matrix of $\Phi$ with respect to the basis $x_{i}$, $i\in I$, is the diagonal matrix ${\rm Diag}(\Phi(x_{i},x_{i}))_{i\in I}$.

The following Decomposition Theorem 2.13 guarantees the existence of orthogonal bases (with respect to Hermitian forms) is the starting point for the classification of Hermitian forms :

2.13 Decomposition Theorem

*Let $K$ be a field with notations and assumptions as in 2.1 and let $\Phi:V\times V\rightarrow K$ be a sesquilinear form on a finite dimensional $K$-vector space $V$. Then in each of the following cases :

(a)  The conjugation of $K$ (see 2.1) is $\neq\operatorname{id}_{K}$ and $\Phi$ is Hermitian or skew-Hermitian,

(b)  ${\rm Char}\,K\neq 2$ and $\Phi$ is a symmetric bilinear form,

$V$ has an orthogonal basis $\mathbcal{x}\!=\!\{x_{1},\ldots,x_{n}\}$, $n\!=\!\operatorname{Dim}_{K}V$, with respect to $\Phi$. In otherwords $:$  $V$ is the orthogonal direct sum V\!=\!\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}_{i=1}^{n}Kx_{i} into $1$-dimensional subspaces $Kx_{i}$, $i\!=\!1,\ldots,n$, with respect to $\Phi$ and \Phi=\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}_{i=1}^{n}\Phi|Kx_{i} is the orthogonal direct sum of its restrictions $\Phi|Kx_{i}$, $i=1,\ldots,n$.

Matrix formulation :  If either $\mathscr{G}\in\operatorname{M}_{I}(K)$, $I$ finite indexed set, is a Hermitain or skew-Hermitian matrix, or if the conjugation is $=\operatorname{id}_{K}$, ${\rm Char}\,K\neq 2$ and $\mathscr{G}$ is a symmetric matrix, then there exists an invertible matrix $\mathscr{A}\in\operatorname{GL}_{I}(K)$ such that ${}^{\rm t}\mathscr{A}\mathscr{G}\overline{\mathscr{A}}$ is a diagonal matrix.*

Proof  Use induction on $\operatorname{Dim}_{K}V$, 2.10.5a, 2.10.5b and 2.10.5c.

2.14

Automorphisms and Congruence*  Let $\Phi:V\times V\rightarrow K$ and $\Psi:W\times W\rightarrow K$ be sesquilinear forms on the $K$-vector spaces $V$ and $W$, respectively. A map $f\,\colon V\rightarrow W$ is called a homomorphism of $(V,\Phi)$ in $(W,\Psi)$ if it is $K$-linear and is compatible with the forms $\Phi$ and $\Psi$, i. e. $\Phi(x,y)=\Psi(f(x),f(y))$ for all $x,y\in V$. A bijective homomorphism $f:(V,\Phi)\to(W,\Psi)$ is called an isomorphism of $(V,\Phi)$ onto $(W,\Psi)$.*

A homomorphism $(V,\Phi)\to(V,\Phi)$ of is called an endomorphism of $(V,\Phi)$ or of $\Phi$. The set of endomorphisms $\operatorname{End}_{K}(V,\Phi)$ (with composition) is a monoid. An isomorphism $(V,\Phi)\rightarrow(V,\Phi)$ is called an automorphism of $(V,\Phi)$ or of $\Phi$. The set of automorphisms $\operatorname{Aut}_{K}(V,\Phi)$ of $(V,\Phi)$ is the unit group of the monoid $\operatorname{End}_{K}(V,\Phi)$ and is called the automorphism group of $(V,\Phi)$ or of $\Phi$ .

If there exists an isomorphism from $(V,\Phi)$ onto $(W,\Psi)$, then $(V,\Phi)$ and $(W,\Psi)$ or also the forms $\Phi$ and $\Psi$ said to be congruent. If $f:(V,\Phi)\rightarrow(W,\Psi)$ is an isomorphism, then the map ${\rm Aut}\,\Phi\to{\rm Aut}\,\Psi$, $g\mapsto fgf^{-1}$ is an isomorphism of groups.

Two square matrices ${\mathscr{C}}$, ${\mathscr{C}}^{\prime}\in{\rm M}_{n}(K)$ are said to be congruent if there exists an invertible matrix ${\mathscr{A}}\in{\rm GL}_{n}(K)$ with ${\mathscr{C}}={}^{\rm t}{\mathscr{A}}\,{\mathscr{C}}^{\prime}\,{{\mathscr{A}}}$.

On sesquilinear forms on finite dimensional $K$-vector spaces (resp. of square matrices over $K$) the relation of “being congruent” is an equivalence relation.

2.14*.1  Let $V$, $W$ be finite dimensional $K$-vector spaces with $\operatorname{Dim}_{K}V\!=\!\operatorname{Dim}_{K}W$, $\mathbcal{x}\!=\!\{x_{i}\mid i\in I\}$, $\mathbcal{y}\!=\!\{y_{i}\mid i\in I\}$ bases of $\,V$, $W$ and let $\Phi$, $\Psi$ be sesquilinear forms on $V$, $W$, resp. Further, let $f:V\rightarrow W$ be an $K$-isomorphism of vector spaces. Then $f$ is an isomorphism $(V,\Phi)\rightarrow(W,\Psi)$ if and only if $\,{\mathscr{G}}_{\Phi}(\mathbcal{x})={}^{\rm t}{\mathscr{M}^{\mathbcal{x}}_{\mathbcal{y}}(f)}\,{\mathscr{G}}_{\Psi}(\mathbcal{y})\,{\overline{\mathscr{M}^{\mathbcal{x}}_{\mathbcal{y}}(f)}}$. In particular, $\Phi$ and $\Psi$ are congruent if and only if there exists $\,{\mathscr{A}}\!\in\!\operatorname{GL}_{I}(K)$ with ${\mathscr{G}}_{\Phi}(\mathbcal{x})={}^{\rm t}{\mathscr{A}}\,{\mathscr{G}}_{\Psi}(\mathbcal{y})\,{\overline{\mathscr{A}}}$.*

2.14*.2  Corollary  Let $\Phi$ be a sesquilinear form on a finite dimensional $K$-vector space $V$ with basis $\mathbcal{x}=\{x_{i}\mid i\in I\}$. Then an automorphism $f\in\operatorname{Aut}_{K}V$ is an automorphism of $(V,\Phi)$ if and only if $\,\mathscr{G}_{\Phi}({\mathbcal{x}})={}^{\rm t}{\mathscr{M}^{\mathbcal{x}}_{\mathbcal{x}}(f)}\,\mathscr{G}_{\Phi}({\mathbcal{x}})\,\overline{\mathscr{M}^{\mathbcal{x}}_{\mathbcal{x}}(f)}$.*

2.15

Classification  problem  for  sesquilinear  forms*  The classification problem for the sesquilinear forms on finite dimensional $K$-vector spaces is to find a well arranged representative system for the equivalence classes of congruent sesquilinear forms.

For example, from the Decomposition Theorem 2.13 it follows immediately that :

2.15.1  Let $K$ be a field with notation and assumptions as in 2.1. Then every pure-Hermitian or pure-skew-Hermitian $($resp. if $\,{\rm Char}\,K\neq 2$, then every symmetric$)$ matrix is congruent to a diagonal matrix $\operatorname{Diag}(c_{i})_{i\in I}\in\operatorname{M}_{I}(K)$, $I$ finite indexed set. Moreover, the form $K^{I}\times K^{I}\rightarrow K$ defined by $(e_{i},e_{j})\mapsto\delta_{ij}c_{i}$, where $e_{i}$, $i\in I$, is the standard basis of $K^{I}\!$ is also congruent to the form defined by $((a_{i}),(b_{i}))\mapsto\sum_{i\in I}a_{i}\overline{b}_{i}c_{i}$.

The above form defined in 2.15.1 (and every other form which is congruent to this form) defined by the diagonal matrix $\operatorname{Diag}(c_{i})_{i\in I}\in\operatorname{M}_{I}(K)$ is denoted by $[c_{i}]_{i\in I}$ and for $I=\{1,\ldots,n\}$ also by $[c_{1},\ldots,c_{n}]$. The form $[c_{i}]_{i\in I}$ is the orthogonal direct sum of the forms $[c_{i}]\,:(a,b)\mapsto a\overline{b}c_{i}$, $i\in I$, on $K$, therefore :  

                                        [c_{i}]_{i\in I}=\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}_{i\in I}\,[c_{i}].*

In general, it is difficult to classify the forms $[c_{i}]_{i\in I}$ up to congruence. Obviously, the form $[c_{i}]_{i\in I}$ is congruent to the form $[a_{i}^{2}c_{i}]_{i\in I}$, where $a_{i}\in K^{\times}\!\!$, $i\in I$, since this is the transition of the basis $e_{i}$, $i\in I$, to the basis $a_{i}e_{i}$, $i\in I$. Therefore, one can replace the elements $c_{i}$ (if non-zero) by their images in the residue class group $\,K^{\times}/{\rm N}(K^{\times}\!)$ of $K^{\times}$ modulo of the subgroup ${\rm N}(K^{\times}\!):=\{a\overline{a}\mid a\in K^{\times}\}$. Note that ${\rm N}(K^{\times}\!)\subseteq K^{\prime\times}\!\!$, where $K^{\prime}$ is the fixed field of the conjugation of $K$ (see 2.1) and if the conjugation is $\operatorname{id}_{K}$, then ${\rm N}(K^{\times}\!)={}^{2}K^{\times}\!$ is the subgroup of the quadratic-units in $K$.

If the form $[c_{i}]_{i\in I}$ is Hermitian, then $c_{i}\in K^{\prime}$ for every $i\in I$ and from the Decomposition Theorem 2.13, we have :

2.16 Theorem

Let $K$ be a field with notation and assumptions as in 2.1 and ${\rm N}(K^{\times}\!)=K^{\prime\times}\!\!$, where $K^{\prime}$ is the fixed field of the conjugation of $K$. Then every Hermitian form of rank $r$ on an $n$-dimensional $K$-vector space is congruent to the form $\,[1,\ldots,1,0,\ldots,0]$, where $1$ occurs $r$ times and [math] occurs $n-r$ times. In particular, every non-degenerate Hermitian form on an $n$-dimensional $K$-vector space is congruent to the standard form $[1,\ldots,1]$ on $K^{n}$.*

2.17 Corollary

Let $K$ be a field with ${\rm Char}\,K\neq 2$ and $K={}^{2}K$ (e. g., if $K$ is algebraically closed, $K=\mathds{C}$). Then all symmetric matrices in ${\rm M}_{n}(K)$ of equal rank are congruent. The diagonal matrices $\,{\rm Diag}\,(0,\ldots,0)$, ${\rm Diag}\,(1,0,\ldots,0)$, $\ldots,$  ${\rm Diag}\,(1,\ldots,1)=\mathscr{E}_{n}\,$ form a complete representative system for the congruence classes of symmetric matrices over $K$.

3. Type and signature of Hermitian forms

In this section, we recall the classification of symmetric and Hermitian forms on finite dimensional vector spaces over a real closed field and its algebraic closure, up to congruence, see Definition 2.10.1. Most of these results can be found in the graduate text books, [18, Ch. V, §12], [17, Ch. IX] or [9, Ch. 11], [1, Ch. 7], or [14, Ch. XV]. However, for the sake of completeness, we recall them without proofs.

3.1

Notation*  (See also 2.1)  Let $K$ be a real closed field 222 Real closed fields  A field $K$ is called real closed if it is real, i. e. if for all $a_{1},\dots,a_{n}\in K$,   $a_{1}^{2}\!+\!\cdots\!+\!a_{n}^{2}\!=\!0$ implies $a_{1}\!=\!\cdots\!=\!a_{n}\!=\!0$. and if it has no nontrivial real algebraic extension $L\,|\,K,$ $L\neq K$. For example, the field $\mathds{R}$ of real numbers is real closed. The algebraic closure of $\mathds{Q}$ in $\mathds{R}$ is real closed. The field $\mathds{Q}$ is real, but not real closed. In 1927 Artin-Schreier proved : A field $K$ is real if and only if there is an order $\leq$ on $K$ such that $(K,\leq)$ is an ordered field. In particular, the characteristic of a real field is [math]. Theorem  (Euler-Lagrange)  Let $(K,\leq)$ be an ordered field satisfying the properties :  (i) Every polynomial $f\in K[X]$ of odd degree has a zero in $K$.  (ii)  Every positive element in $K$ is a square in $K$. Then the field $\overline{K}=K({\rm i})$ obtained from $K$ by adjoining a square root ${\rm i}$ of $-1$ is algebraically closed. In particular, $K$ itself is real-closed. For a proof see [9, Ch. 11, §11.1]. (Remark : Since the field $\mathds{R}$ of real numbers is ordered and satisfies the properties (i) and (ii), the Eulae-Lagrange theorem proves the Fundamental Theorem of Algebra : The field $\mathds{C}=\mathds{R}({\rm i})$ of complex numbers is algebraically closed . The Euler-Lagrange Theorem has a remarkable complement : — Theorem  (Artin - Schreier)  Let $L$ be an algebraically closed field. If $K\subseteq L$ be a subfield of $L$ such that $\,L\,|\,K$ is finite and $K\neq L$, then $L=K({\rm i})$ with ${\rm i}^{2}+1=0$ and $K$ is a real-closed field. For a proof see [9, Ch. 11, §11.7].). Then $\operatorname{Aut}K=\{\operatorname{id}_{K}\}$ and the field $\mathds{C}_{K}:=K[{\rm i}\,]$, where ${\rm i}^{2}\!=-1$, of (complex) numbers over $K$, is the algebraic closure of $K$ with the Galois group $\operatorname{Gal}(\mathds{C}_{K}\,|\,K)=\{\operatorname{id}_{\mathds{C}_{K}},\kappa\}$, where $\kappa:\mathds{C}_{K}\to\mathds{C}_{K}$, is the (complex)-conjugation defined by by ${\rm i}\mapsto-{\rm i}\,$, see 2.1 and Examples 2.2.

Further, we denote by $\mathds{K}$ either the field $K$, or  the field $\mathds{C}_{K}$. The involution in the case $\mathds{K}=K$ is $\operatorname{id}_{K}$ and in the case $\mathds{K}=\mathds{C}_{K}$ is the (complex)-conjugation $\kappa:\mathds{C}_{K}\to\mathds{C}_{K}$ which we will simply denote by the standard bar-notation, i. e. $a\mapsto\overline{a}$, $a\in\mathds{C}_{K}$.

With these notation and assumptions, we note that the term “Hermitian ” means “real-symmetric” in the case $\mathds{K}=K$ and “complex-Hermitian” if $\mathds{K}=\mathds{C}_{K}$.*

3.2 Proposition

*With the notation as in 3.1, let $\Phi:V\times V\longrightarrow\mathds{K}$ be a Hermitian form on the finite dimensional $\mathds{K}$-vector space $V$. Then there exists an orthogonal basis $\mathbcal{x}=\{x_{1},\ldots,x_{n}\}$, $n:=\operatorname{Dim}_{K}V$ of $V$ with respect to $\Phi$ such that the Gram’s matrix $\mathscr{G}_{\Phi}(\mathbcal{x})$ is a diagonal matrix :

                           $\,\mathscr{E}_{n}^{p,q}:={\rm Diag}\,(\underbrace{1,\ldots,1}_{p\,\hbox{\scriptsize-times}},\underbrace{-1,\ldots,-1}_{q\,\hbox{\scriptsize-times}},\underbrace{0,\ldots,0}_{n-p-q\,\hbox{\scriptsize-times}})$.*

Below in 3.4 we note that that the numbers $p$ and $q$ are uniquely determined by $\Phi$. Obviously, $p+q=\operatorname{rank}\,\Phi$. In particular, $\Phi$ is non-degenerate if and only if $p+q=n$. For characterization of invariants $p$ and $q$ the following concepts are useful :

3.3 Definition

Let $\Phi:V\times V\longrightarrow\mathds{K}$ be a Hermitian form on the finite dimensional $\mathds{K}$-vector space $V$. Then $\,\Phi$ is called :

(1) positive definite if $\Phi(x,x)>0$ for all $x\in V$, $x\neq 0$.

(2) negative definite if $\Phi(x,x)<0$ for all $x\in V$, $x\neq 0$.

(3) positive semi - definite if $\Phi(x,x)\geq 0$ for all $x\in V$.

(4) negative semi - definite if $\Phi(x,x)\leq 0$ for all $x\in V$.

(5) indefinite if there are vectors $x,y\in V$ with $\Phi(x,x)>0$ and $\Phi(y,y)<0$.

3.4

Sylvester’s Law of Inertia*  Let $\Phi$ be a Hermitian form on the finite dimensional $\mathds{K}$-vector space $V$ and let $\mathbcal{x}=\{x_{1},\ldots,x_{n}\}$, $n:=\operatorname{Dim}_{\mathds{K}}V$ be an orthogonal basis of $V$ with respect to $\Phi$ such that the Gram’s matrix $\mathscr{G}_{\Phi}(\mathbcal{x})$ of $\Phi$ is the diagonal matrix :

                           $\,\mathscr{E}_{n}^{p,q}:={\rm Diag}\,(\underbrace{1,\ldots,1}_{p\,\hbox{\scriptsize-times}},\underbrace{-1,\ldots,-1}_{q\,\hbox{\scriptsize-times}},\underbrace{0,\ldots,0}_{n-p-q\,\hbox{\scriptsize-times}})$.

Then $p$ is the maximum of the dimensions of subspaces of $V$ on which $\Phi$ is positive definite, and $q$ the maximum of the dimensions of subspaces of $V$ on which $\Phi$ is negative definite. — In particular, $p$ and $q$ do not depend on the special choice of the orthogonal basis $x_{1},\ldots,x_{n}$ of $\,V$.*

3.5 Definition

The pair $(p,q)$ as in the Sylvester’s Law of Inertia 3.4 is called the type of the form $\Phi$. The natural number $p$ is called the (inertia -) index, the integer $p-q$ is called the signature and the natural number $q$ is called the Morse - index of $\Phi$. We denote the rank, signature and type of a Hermitian form $\Phi$ by $\operatorname{rank}\,\Phi$, $\operatorname{sign}\,\Phi$ and $\operatorname{type}\,\Phi$, resp.

The type of a Hermitian matrix $\,\mathscr{C}\in{\rm M}_{n}(\mathds{K})$ is by definition the type a form with $\mathscr{C}$ as the Gram’s matrix with respect to an (arbitrary) $\mathds{K}$-basis of $\mathds{K}^{n}$. The matrix analog of the Sylvester’s Law of Inertia 3.4 is the following :

3.6 Corollary

Let $\Phi$ be a Hermitian form on the $n$-dimensional $\mathds{K}$-vector space $V$ with $\mathds{K}$-basis $\mathbcal{x}=\{x_{1},\ldots,x_{n}\}$. Then $\Phi$ is of type $(p,q)$ if and only if the Gram’s matrix $\mathscr{G}_{\Phi}(\mathbcal{x})$ is congruent to the matrix $\mathscr{E}_{n}^{p,q}$, i. e. there exists an invertible matrix $\mathscr{A}\in{\rm GL}_{n}\,(\mathds{K})$ such that $\mathscr{G}_{\Phi}({\mathbcal{x}})=\,^{\rm t}\mathscr{A}\mathscr{E}_{n}^{p,q}\,\overline{\mathscr{A}}\,$. Two Hermitian matrices $\mathscr{C}$ and $\mathscr{C}^{\prime}\in{\rm M}_{n}(\mathds{K})$ have the same type if and only if they are congruent. In particular, a Hermitian matrix $\mathscr{C}\in{\rm M}_{n}(\mathds{K})$ have type $(p,q)$ if and only if $\mathscr{C}$ is congruent to the matrix $\mathscr{E}_{n}^{p,q}$.

If $\mathds{K}\!=\!K$ (real closed), then one can choose 333 Use the following observation : Let $V$ be an oriented vector space over a real-closed field $K$ of dimension $n\in\mathds{N}^{*}$ and $\Phi$ be a Hermitian form of type $(p,q)$ on $V$. Then there exists an orientation of $V$ represented by a basis $x_{1},\ldots,x_{n}$ of $V\,$ such that the Gram’s matrix of $\Phi$ is equal to the matrix ${\mathscr{E}}_{n}^{p,q}$.  $\mathscr{A}\!\in\!{\rm GL}_{n}^{+}(K)$, i. e. $\operatorname{Det}\mathscr{A}>0$. In the situation of Corollary 3.6, if $\Phi$ is non-degenerate, i. e. if $p+q\!=\!n$, then ${\rm Det}\,{\mathscr{G}}_{\Phi}({\mathbcal{x}})=(-1)^{q}|\,{\rm Det}\,{\mathscr{A}}|^{2}$, i. e. ${\rm Sign}\,({\rm Det}\,{\mathscr{G}}_{\Phi}({\mathbcal{x}}))\!=\!(-1)^{q}\!$. Therefore, the sign of the Gram’s determinant ${\rm Det}\,{\mathscr{G}}_{\Phi}({\mathbcal{x}})\,$ determines the parity of $q$. From this the following useful criterion for the determination of the type follows :

3.7

Hurwitz’s Criterion*  Let $\Phi$ be a Hermitian form on the $n$-dimensional $\mathds{K}$-vector space $V$ with basis $\mathbcal{x}=\{x_{1},\ldots,x_{n}\}$. Suppose that the principal minors*

[TABLE]

of the Gram’s matrix $\mathscr{G}_{\Phi}({\mathbcal{x}})=(\Phi(x_{i},x_{j}))\in{\rm M}_{n}\,(K)$ of $\Phi$ with respect to the basis $\mathbcal{x}$ are all $\neq 0$. Then the type of $\Phi$ is $(n-q,q)$, where $q$ is the number of sign changes 444 Recall that we say that in a sequence $a_{0},\ldots,a_{n}$ of non-zero real numbers changes the sign at the $i$-th place if $\,0\leq i<n\,$ and $\,a_{i}\,a_{i+1}<0$. — For an arbitrary sequence of real numbers $b_{0},\ldots,b_{m}$ by a change of signs means a change of signs in the sequence obtained by removing the zeros from the original sequence. in the sequence $1=D_{0},D_{1},\ldots,D_{n}={\rm Det}\,{\mathscr{G}_{\Phi}({\mathbcal{x}})}$.

3.8 Corollary

Let $\Phi$ be a Hermitian form on the $n$-dimensional $\mathds{K}$-vector space $V$ with basis $\mathbcal{x}=\{x_{1},\ldots,x_{n}\}$ and let

[TABLE]

be the principal minors of the Gram’s matrix $\mathscr{G}_{\Phi}(\mathbcal{x})$. Then :

$(1)$*  $\Phi$ is positive definite if and only if $\,D_{i}>0$  for all $\,i=1,\ldots,n$.*

($2)$  $\Phi$ is negative definite if and only if $(-1)^{i}\,D_{i}>0\,$ for all $i=1,\ldots,n$,  i. e. at every position in the sequence $D_{0},D_{1},\ldots,D_{n}$ there is a sign change.

3.9 Example

*Let $\{v_{1},v_{2}\}$ be a basis of the 2-dimensional $K$-vector space $V$. For the symmetric bilinear form $\Phi\!=\!\langle-,-\rangle$ on $V$. Let $D_{1}\!=\!\langle v_{1},v_{1}\rangle$ and $D_{2}\!=\!{\rm Det}\begin{pmatrix}\langle v_{1},v_{1}\rangle&\langle v_{1},v_{2}\rangle\cr\langle v_{2},v_{1}\rangle&\langle v_{2},v_{2}\rangle\cr\end{pmatrix}\!=\!\langle v_{1},v_{1}\rangle\,\langle v_{2},v_{2}\rangle-|\langle v_{1},v_{2}\rangle|^{2}$. Then the following table shows the dependence of the ${\rm sign}\,D_{1}$, ${\rm sign}\,D_{2}$ and the type of $\Phi$ :

          \offinterlineskip\vbox{\halign{\strut\ # &#\vrule&\ #\ &#\vrule&\ #\ &#\vrule&\ #\ &#\vrule&\ #\ &#\vrule&\ #\ &#\vrule&\ #\ &#\vrule&\ #\ &#\vrule&\ #\ & \ #\vrule& \ # & \ # \vrule& \ # & \ # \vrule&\ #\ \cr D_{1}&&+&&+&&--&&--&&+&&--&&0&&0 &&0 &&0 \cr\hrule\cr D_{2}&&+&&--&&+&&--&&0&&0&&--&&0 &&0 &&0 \cr\hrule\cr\langle v_{1},v_{2}\rangle&&&&&&&&&&&&&&&&>\,0 &&<\,0 &&0 \cr\hrule\cr{\rm Type}&&\ (2,0) \ &&\ (1,1) \ &&\ (0,2) \ &&\ (1,1) \ &&\ (1,0) \ &&\ (0,1) \ &&\ (1,1) \ &&\ (1,0)\\ &&\ (0,1) \ &&\ (0,0) \ \cr}}\ \,.

Note that the case $D_{1}=0$, $D_{2}>0$ is not possible.*

3.10 Example

Let $z\in\mathds{C}_{K}\smallsetminus K$, $\pi:=(X-z)(X-\overline{z})\in K[X]$, $A:=K[X]/\langle\pi\rangle]:=K[x]$, where $x$ is the image of $X$ modulo $\langle\pi\rangle$. Further, let $H\in K[X]$, $H\not\in\langle\pi\rangle$, $h=h(x)\in A$ be the image of $H$ in $A$ and let $\Phi_{h}:A\times A\longrightarrow K$ be the symmetric bilinear form defined by $\Phi_{h}(f,g)=\operatorname{Tr}_{K}^{A}(hfg)$, $\,f$, $g\in A$. Then the Gram’s matrix of $\Phi_{h}$ with respect to the basis $\{1,x\}$

[TABLE]

is a symmetric matrix with $\,D_{1}=h(z)+h(\overline{z})=2\,{\rm Re}\,h(z)\,$ and $\,D_{2}=\operatorname{Det}\,\mathscr{G}_{\Phi_{h}}(1,x)=h(z)\,h(\overline{z})\,(z-\overline{z})^{2}=-4\,|h(z)|^{2}({\rm Re}\,z\,)^{2}<0$. Therefore, by the table in Example 3.10, the type of $\Phi_{h}$ is $(1,1)$

The type of a Hermitian form on a finite dimensional vector space $V$ over $\mathds{C}_{K}$ can also be determined by using the eigenvalues of the Gram’s matrix, see Theorem 3.12 below. Usual proofs given in the standard text books of this fact uses the Principal Axis Theorem for self-adjoint operators (also known ans the Spectral Theorem). We give here a direct proof using the following interesting Lemma 3.11 :

3.11 Lemma

Let $K$ be a real closed field with notation as in 3.1, $V$ an $n$-dimensional $\mathds{C}_{K}$-vector space $V$ with a positive definite complex-Hermitian form $\Phi$ on $V$ and let $f:V\rightarrow V$. Then there exists an orthonormal basis $\mathbcal{x}\!=\!(x_{1},\ldots,x_{n})$ of $V$ such that the matrix $\mathscr{M}^{\mathbcal{x}}_{\mathbcal{x}}(f)$ of $f$ with respect to $\mathbcal{x}$ is an upper triangular matrix.

3.12 Theorem

Let $K$ be a real closed field with notation as in 3.1 and let ${\mathscr{C}}\in\operatorname{M}_{n}(\mathds{C}_{K})$ be a Hermitian matrix. Then all the eigenvalues of ${\mathscr{C}}$ are in $K$ and ${\mathscr{C}}$ is of type $(p,q)$, where $p$ is the number of positive eigenvalues and $q$ is the number of negative eigenvalues of ${\mathscr{C}}$ counted with their multiplicities in the characteristic polynomial $\chi_{\mathscr{C}}$ of ${\mathscr{C}}$.

3.13 Remark

The proof of the above Theorem 3.12 also shows that every Hermitian matrix in $\operatorname{M}_{n}(\mathds{C}_{K})$ is diagonalizable (even with respect to an orthonormal basis of $\mathds{C}_{K}^{n}$).

3.14 Corollary

Let $K$ be a real closed field with notation as in 3.1 and let ${\mathscr{C}}\in\operatorname{M}_{n}(\mathds{C}_{K})$ be a Hermitian matrix. Then the characteristic polynomial $\chi_{\mathscr{C}}=c_{0}+c_{1}X+\cdots+c_{n-1}X^{n-1}+X^{n}\in K[X]$ and ${\mathscr{C}}$ is of type $(p,q)$, where $p$ is the number of sign changes in the sequence $c_{0},c_{1},\ldots,c_{n-1},c_{n}:=1$ and $q$ is the number of sign changes in the sequence $c_{0},-c_{1},\ldots,(-1)^{n-1}c_{n-1},(-1)^{n}c_{n}=(-1)^{n}$. If $c_{0}=c_{1}=\cdots=c_{r-1}=0$ and $c_{r}\neq 0$, then $p+q=n-r$.

Proof  Note that, since all the eigenvalues of ${\mathscr{C}}$ are real by Theorem 3.12, indeed $\chi_{\mathscr{A}}\in K[X]$. The assertion is immediate from Theorem 3.12 and the following classical theorem of Descartes :

3.15 Theorem

(Descartes’ Rule of Signs)*  Let $K$ be a real closed field with notation as in 3.1 and let $f=a_{0}+a_{1}X+\cdots+a_{n-1}X^{n-1}+a_{n}X^{n}\in K[X]$, $a_{n}\neq 0$ be a polynomial of degree $n$. Further, let ${\rm V}_{+}$, $($resp. ${\rm V}_{-})$ denote the number of sign changes in the sequence $a_{0},a_{1},\ldots,a_{n-1},a_{n}$ *(resp. in the sequence $a_{0},-a_{1},\ldots,(-1)^{n-1}a_{n-1},(-1)^{n}a_{n})$ and $N_{+}$ $($resp. $N_{-})$ denote the number of positive $($resp. negative$)$ zeros of $f$ $($each zero of $f$ is counted with its multiplicity$)$. Then there exist natural numbers $r_{+}$ and $r_{-}\in\mathds{N}$ such that $N_{+}={\rm V}_{+}-2r_{+}$ and $N_{-}={\rm V}_{-}-2r_{-}$. Moreover, if all zeros of $f$ belong to $K$, i. e. if $f$ splits into linear factors in $K[X]$, then $N_{+}={\rm V}_{+}$ and $N_{-}={\rm V}_{-}$.

We now recall (from [3]) that “being of type $(p,q)$” is an open property (with respect to the *strong topology 555 Strong topology  Let $K$ be a real closed field (see Footnote 2). Then $K$ is equipped with the order topology which is determined by the base of the open intervals $]\,a,b\,[$, $a,b\in K$, $a<b$. The $K$-vector spaces $K^{n}$, $n\in\mathds{N}$, are endowed with the product topology (with the base given by the open cuboids $]\,a_{1},b_{1}\,[\times\cdots\times]\,a_{n},b_{n}\,[$, $a_{i}<b_{i}$, $i=1,\ldots,n$). With the ordered and product topology, the addition, the multiplication and the inverse are continuous functions on $K\times K$ and $K^{\times}=K\setminus\{0\}$, respectively. Further, polynomial functions (resp. rational functions $F/G$, $F$, $G\in K[X_{1},\ldots,X_{n}]$, $G\neq 0$), in $n$ variables are continuous $K$-valued functions on $K^{n}$ (resp. on $K^{n}\smallsetminus{\rm V}_{K}(G)$, where ${\rm V}_{K}(G):=\{a\in K\mid G(a)=0\}$ is a zero set of the denominator $G$ which is closed in $K^{n}$.

The product topology on $K^{n}$ transfers uniquely to every $n$-dimensional $K$-vector space by a $K$-linear isomorphism $f:V\rightarrow K^{n}\!\!$. Any other isomorphism $g:V\rightarrow K^{n}$ defines the same topology, since $gf^{-1}:K^{n}\rightarrow K^{n}$ and $(gf^{-1})^{-1}=fg^{-1}:K^{n}\rightarrow K^{n}$ are continuous (polynomial) maps. Therefore, polynomial and rational functions are also defined on any finite dimensional vector space $V$ by an isomorphism $f:V\rightarrow K^{n}\!\!$. This topology on $V$ may be characterized as the smallest topology for which the $K$-linear functions $V\rightarrow K$ are continuous and is called the strong topology on $V$, since it is stronger than the Zariski topology on $V$ if $V\neq 0$.* ) which is an easy consequence of Hurwitz’s Criterion 3.7 :

3.16 Lemma

(cf. [3, Lemma 1.2]* Let $K$ be a real closed field with notations as in 3.1 and $F_{ij}\in K[T]$ be polynomials such that $F_{ij}\!=\!F_{j\,i}$, $1\leq i,j\leq n$. Suppose that the bilinear form defined by the symmetric matrix $(F_{ij}(s))_{1\leq i,j\leq n}\!\in\!\operatorname{M}_{n}(K)$ at $s\!\in\!K$, is non-degenerate, then there exists an $\varepsilon>0$ such that the type of the symmetric matrices $\,(F_{ij}(t))_{1\leq i,j\leq n}$ is the same for all $t\in]\,s-\varepsilon,s+\varepsilon\,[$. In particular, for non-degenerate symmetric bilinear forms over $K$, “being of type $(p,q)$” is an open property.*

We end this section by noting the following important Rigidity Theorem for symmetric bilinear forms (see [3] which is proved by using Hurwitz’s Criterion 3.7, the above Lemma 3.16 and the Intermediate Value Theorem for polynomial functions 666 Intermediate Value Theorem for polynomial functions  Let $K$ be a real closed field and $F\in K[T]$ be a polynomial with coefficients in $K$ such that $F(a)F(b)<0$ for some $a,b\in K$. Then $F$ has a zero in $[a,b]$. In other words, the values $F(t)$, $t\in[a,b]$, have the same sign if $F$ has no zero on $[a,b]$. In particular, every polynomial of odd degree has a zero in $K$. A field with this property is called a $2$-field. Therefore, a real closed field is a $2$-field. Furthermore, every monic polynomial $F$ over a real closed field $K$ has a positive zero in $K$ if $F(0)<0$ (since $F(x)>0$ for “large” $x$). .

3.17

Rigidity Theorem for Quadratic Forms* (cf. [3, 1.3])  Let $K$ be a real closed field with notations as in 3.1 and let $R_{ij}(t)=R_{ij}(t_{1},\ldots,t_{n})$, $1\leq i\,,\,j\leq n$, be rational functions on a line-connected 777 Line-connected subsets  Let $V$ be a vector space over a real closed field $K$. For two points $x,y\in V$, the subset $[x,y]=[y,x]:=\{(1-t)x+ty\mid t\in K,0\leq t\leq 1\}\subseteq V$ is called the (closed) line-segment connecting $x$ and $y$. For $x_{0},\ldots,x_{r}\in V$, $r\geq 1$, the subset $[x_{1},\ldots,x_{r}]:=\cup_{i=1}^{r}[x_{i-1},x_{i}]$ is called the broken line from $x_{0}$ to $x_{r}$. A subset $V^{\prime}\subseteq V$ is called line-connected if for any two points $x,y\in V^{\prime}$ there is a broken line from $x$ to $y$ which lies entirely in $V^{\prime}$. Note that, if $K=\mathds{R}$ and $U\subseteq V$ is open (in the strong topology, see Footnote 5), then the notion “line-connected” is equivalent to the topological notion of “connected”. The only topologically connected subspaces of $K=\mathds{Q}$ are the singletons. If $V$ is a line, i. e. $1$-dimensional, and if $x\in V$, then $V\smallsetminus\{x\}$ is not line-connected. However, if $\,\operatorname{Dim}_{K}\!V\geq 2$, then $V\smallsetminus\{x\}$ is always line-connected : If $u,w\in V\smallsetminus\{x\}$ are arbitrary points, there is always a point $v\in V\setminus\{x\}$ such that $[u,v,w]\subseteq V\smallsetminus\{x\}$. subset $U\subseteq K^{n}$ such that the matrices $\mathscr{R}(t)=(R_{ij}(t))_{1\leq i,j\leq n}\in\operatorname{M}_{n}(K)$, $t\in U$, are symmetric, i. e. $R_{ij}=R_{ji}$ for all $1\leq i\,,\,j\leq n$ with a $\operatorname{Det}\,\mathscr{R}(t)\neq 0$ for all $t\in U$. Then all the matrices $\mathscr{R}(t)\in\operatorname{M}_{n}(K)$, $t\in U$, have the same type $\,(p,q)$, or equivalently, the same signature $p-q$.*

4. Trace forms and Rational points

In this section, we recall the results from [3] (based on the talk of Prof. U. Storch at IIT Bombay in November 2009) on trace forms, their invariants such as rank, type, signature and their relations with the number of rational points of a finite algebra $A$ over a real closed field. For detailed proofs of these results the reader is recommended to see [3, $\S$ 3].

4.1

Preliminaries *  In this subsection, we recall the basic concepts from elementary commutative algebra (see [2], [12] and [15]).*

*Let $A$ be an arbitrary commutative ring (with unity). The set $\mathrm{Spec}A$ (resp. $\operatorname{Spm}A$) of prime (resp. maximal) ideals in $A$ is called the prime (resp. maximal) spectrum of $A$. The nil-radical $\mathfrak{n}_{A}:=\sqrt{0}=\cap_{\,\mathfrak{p}\in\mathrm{Spec}A}\,\mathfrak{p}$ is the intersection of all prime ideals in $A$. More generally, (formal Nullstellensatz)  $\sqrt{\mathscr{A}}=\cap_{\,\mathfrak{p}\in\mathrm{Spec}A}\,\{\mathfrak{p}\mid\mathscr{A}\subseteq\mathfrak{p}\}$ for every ideal $\mathscr{A}$ in $A$, see [2], [12].

The intersection $\displaystyle\mathscr{M}_{A}:=\cap_{\,\mathscr{M}\in\operatorname{Spm}A}\,\mathscr{M}$ of all maximal ideals in $A$ is called the Jacobson radical of $A$.*

• (a)

The ${\bf K}$-Spectrum and the set of ${\bf K}$-rational points of a ${\bf K}$-algebra*  (see [15]) Let $K$ be a field. Using the universal property of the polynomial algebra $K[X_{1},\ldots,X_{n}]$, the affine space $K^{n}$ can be identified with the set of $K$-algebra homomorphisms $\mathrm{Hom}_{{\mathop{K\hbox{{\rm-}}{\rm alg}}\nolimits}}(K[X_{1},\ldots,X_{n}]\,,K)$ by identifying the point $a=(a_{1},\ldots,a_{n})\in K^{n}$ with the substitution homomorphism $\xi_{a}:K[X_{1},\ldots,X_{n}]\rightarrow K$, $X_{i}\mapsto a_{i}$, whose kernel $\operatorname{Ker}\,\xi_{a}$ is the maximal ideal $\mathfrak{m}_{a}=\langle X_{1}-a_{1},\ldots,X_{n}-a_{n}\rangle$ in $K[X_{1},\ldots,X_{n}]$. Moreover, every maximal ideal $\mathfrak{m}$ in $K[X_{1},\ldots,X_{n}]$ with $K[X_{1},\ldots,X_{n}]/\mathfrak{m}=K$ is of the type $\mathfrak{m}_{a}$ for a unique point $a=(a_{1},\ldots,a_{n})\in K^{n}$; the component $a_{\,i}$ is determined by the congruence $X_{i}\equiv~{}a_{i}~{}{\rm mod}~{}\mathfrak{m}$.*

The subset $\,K\operatorname{\!-Spec}K[X_{1},\ldots,X_{n}]:=\{\mathfrak{m}_{a}\mid a\in K^{n}\}$ of $\operatorname{Spm}K[X_{1},\ldots,X_{n}]$ is called the $K$-spectrum of $K[X_{1},\ldots,X_{n}]$. We have the identifications :

[TABLE]

*More generally, for any $K$-algebra $A$, the map *

                  $\,\mathrm{Hom}_{{\mathop{K\hbox{{\rm-}}{\rm alg}}\nolimits}}(A\,,K)\longrightarrow\{\mathscr{M}\in\operatorname{Spm}A\mid A/\mathscr{M}=K\}$, $\xi\longmapsto\operatorname{Ker}\xi$, *

is bijective. Therefore we make the following definition :*

*For any $K$-algebra $A$ of finite type, the subset $\,K\operatorname{\!-Spec}A:=\{\mathscr{M}\in\operatorname{Spm}A\mid A/\mathscr{M}=K\}\,$ is called the $K$- spectrum of $A$ and is denoted by $K\operatorname{\!-Spec}A$. *

Further, if A\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}K[X_{1},\ldots,X_{n}]/\mathfrak{A} is a representation of the finite $K$-algebra $A$, then the $K$-algebraic set ${\rm V}_{K}(\mathfrak{A}:=\{a\in K^{n}\mid F(a)=0\,\ \hbox{for all}\ \,F\in\mathfrak{A}\}$ defined by the ideal $\mathfrak{A}$ is called the set of $K$- rational points of $A$.*

Under the above bijective maps, we have the identification ${\rm V}_{K}(\mathfrak{A})=\mathrm{Hom}_{{\mathop{K\hbox{{\rm-}}{\rm alg}}\nolimits}}(A\,,K)=K\operatorname{\!-Spec}A$. For example, since $\mathds{C}$ is an algebraically closed field, $\operatorname{Spm}\mathds{C}[X]={\mathop{\mathds{C}\hbox{{\rm-}}{\rm Spec}\,}\nolimits}\mathds{C}[X]$, but ${\mathop{\mathds{R}\hbox{{\rm-}}{\rm Spec}\,}\nolimits}\mathds{R}[X]\subsetneqq\operatorname{Spm}\mathds{R}[X]$. In fact, the maximal ideal $\mathfrak{m}:=\langle X^{2}+1\rangle\in\operatorname{Spm}\mathds{R}[X]$ does not belong to ${\mathop{\mathds{R}\hbox{{\rm-}}{\rm Spec}\,}\nolimits}\mathds{R}[X]$. More generally, a field $K$ is algebraically closed if and only if $\operatorname{Spm}K[X]=K\operatorname{\!-Spec}K[X]$, see  [2] , [12] or [6, Theorem 2.10 , HNS 3] )*

• (b)

Local components of a finite algebra*  Let $A$ be a finite algebra over a field $K$, i. e. $A$ finite dimensional as a $K$-vector space of dimension ${\rm Dim}_{K}A$. Then $\operatorname{Spm}A=\mathrm{Spec}A$ (since any finite $K$-algebra which is an integral domain is already a field). Moreover, from the Chinese Remainder Theorem, it follows immediately that $\operatorname{Spm}A$ is a finite set. In particular, $\#\operatorname{Spm}A\leq\operatorname{Dim}_{K}\!A$ and equality holds if and only if $A$ is isomorphic to the product $K$-algebra $K^{\,\operatorname{Dim}_{K}\!A}$ ). *

Further, let $\operatorname{Spm}A=\{\mathscr{M}_{1},\ldots,\mathscr{M}_{r}\}$. Then the unit group $A^{\times}$ of $A$ is $\,A\!\smallsetminus\,\bigcup_{\,i=1}^{\,r}\mathscr{M}_{i}$ and the canonical homomorphism $A\longrightarrow\prod_{\,i=1}^{\,r}A_{\mathscr{M}_{i}}$ is injective (where $A_{\mathfrak{p}}$ denotes the localization of $A$ at a prime ideal $\mathfrak{p}\in\mathrm{Spec}A$). In our special case, it is also surjective and hence an isomorphism, cf. [17, Corollary 55.16]. Therefore, $A$ is the direct product of the local finite $K$-algebras $A_{i}:=A_{\mathscr{M}_{i}}$, $i=1,\ldots,r$, which are called the local components of $A$. Furthermore, we have :  $\,\operatorname{Dim}_{K}A=\sum_{\,i=1}^{r}\operatorname{Dim}_{K}A_{i}=\sum_{i=1}^{r}\ell(A_{i})\cdot[K_{i}:K]$, where, for $\,i=1,\ldots,r$, $K_{i}=A/\mathscr{M}_{i}$ is the residue class field of $A$ at $\mathscr{M}_{i}$ and $\ell(A_{i})$ the (finite) length of $A_{i}$, i. e. the length $\ell$ of a composition series $0=\mathscr{A}_{0}\subsetneq\mathscr{A}_{1}\subsetneq\cdots\subsetneq\mathscr{A}_{\ell}=A_{i}$ with $\mathscr{A}_{j+1}/\mathscr{A}_{j}\cong A/\mathscr{M}_{i}$, $j=1,\ldots,\ell-1$.*

For example, if $K$ is a $2$-field, then $\,[K_{i}:K]\,$ is even if $K_{i}$ is a non-trivial field extension of $K$ and, in particular, $K\operatorname{\!-Spec}A\neq\emptyset$ if $\,\operatorname{Dim}_{K}A$ is odd.*

Further, $\mathscr{M}_{A}=\mathscr{M}_{1}\cap\cdots\cap\mathscr{M}_{r}=\cap_{\mathfrak{p}\in\mathrm{Spec}A}\,\mathfrak{p}=\mathfrak{n}_{A}$ and $\mathscr{M}_{A}=\mathfrak{n}_{A}=0$, i. e. $A$ is reduced if and only if $A=K_{1}\times\cdots\times K_{r}$ is the product of its residue class fields. Moreover, if all the field extensions $K_{i}$ of $K$ are separable, then $A$ is called a (finite) separable $K$-algebra.

4.2

The trace form*  Let $A$ be a finite algebra over the field $K$. The trace form on $A$ over $K$ is the symmetric $K$-bilinear form $\operatorname{Tr}:=\operatorname{Tr}_{K}^{A}:A\times A\rightarrow K$, $(f,g)\mapsto\operatorname{Tr}_{K}^{A}(fg)$ on $A$. It is a classical tool used to study the $K$-algebra $A$.

The decomposition of $A=A_{1}\times\cdots\times A_{r}$ into its local components (cf. 4.1 (c)) yields the orthogonal decomposition (see Decompsition Theorem 2.13)

                                     \,\operatorname{Tr}_{K}^{A}=\operatorname{Tr}_{K}^{A_{1}}\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}\cdots\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}\operatorname{Tr}_{K}^{A_{r}}\,

of the trace form. The degeneration space $A^{{}^{{}_{\bot}}}\!\!\!=\!A^{{}^{{}_{\bot_{\operatorname{Tr}}}}}\!\!\!=\!\{f\in A\mid\operatorname{Tr}(Af)\!\!=\!0\}$ is an ideal in $A$.*

4.3 Lemma

(cf. [3, Lemma 3.1])*  Let $A$ be a finite algebra over an arbitrary field $K$ and let $A^{{}^{{}_{\bot}}}$ be the degeneration space of the trace form $\operatorname{Tr}_{K}^{A}$. Then radical $\mathscr{M}_{A}=\mathfrak{n}_{A}\subseteq A^{{}^{{}_{\bot}}}$. Moreover, equality holds if and only if all the residue class fields of $A$ are separable over $K$, i. e. if and only if the reduction $A_{\operatorname{red}}=A/\mathfrak{n}_{A}$ is a separable $K$-algebra. — In particular, the trace form is non-degenerate if and only if $A$ is a separable $K$-algebra.*

4.4 Corollary

Let $A$ be a finite separable algebra over an arbitrary field $K$. Then

$\,\displaystyle\operatorname{rank}\,\operatorname{Tr}_{K}^{A}=\operatorname{Dim}_{K}(A/\mathscr{M}_{A})=\sum_{i=1}^{r}\,[K_{i}:K]$.

Moreover, if $K$ is an ordered field, then :

$\displaystyle\operatorname{type}\,\operatorname{Tr}_{K}^{A}=\sum_{i=1}^{r}\operatorname{type}\operatorname{Tr}_{K}^{K_{i}}\,$*  and   $\,\displaystyle\operatorname{sign}\operatorname{Tr}_{K}^{A}=\sum_{i=1}^{r}\operatorname{sign}\,\operatorname{Tr}_{K}^{K_{i}}$.*

Now, we state the following important and classical criterion for the existence of $K$-rational points for real closed fields which is proved in [3].

4.5 Theorem

(cf. [3, Theorem 3.2])*  Let $A$ be a finite algebra over a real closed field $K$. Then :*

$\,\operatorname{sign}\,\operatorname{Tr}_{K}^{A}=\#\,K\operatorname{\!-Spec}A$.

In particular, $K$ is a residue class field of $A$ if and only if $\,\operatorname{sign}\,\,\operatorname{Tr}_{K}^{A}\neq 0$.

4.6 Example

Let $K$ be a real closed field. Then there is a unique (up to isomorphism) non-trivial finite field extension $L\,|\,K$, namely, the quadratic field $L=\mathds{C}_{K}=K[\,{\rm i}\,]$ with ${\rm i}^{2}=-1$, of complex numbers over $K$ (which is the algebraic closure of $K$). The Gram’s matrix of the trace form ${\operatorname{Tr}}^{\mathds{C}_{K}}_{K}$ of $\mathds{C}_{K}$ over $K$ with respect to the basis $1$, ${\rm i}\,$ is the matrix

[TABLE]

Therefore $\,\operatorname{type}\,\operatorname{Tr}_{K}^{\mathds{C}_{K}}=(1,1)\,$ and $\,\operatorname{sign}\,\operatorname{Tr}_{K}^{\mathds{C}_{K}}=0$.

4.7 Corollary

Let $A$ be a finite algebra over a real closed field $K$. Then the trace form $\operatorname{Tr}_{K}^{A}$ is positive definite if and only if $A$ is separable over $K$ and $A$ splits over $K$, i. e. there exists an isomorphism of $K$-algebras A\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}K^{\,\operatorname{Dim}_{K}A}.

4.8 Corollary

Let $K$ be a real closed field and $f\in K[X]$ be a monic polynomial. Then all zeros of $f$ (in $\overline{K}$) belong to $K$ and are simple if and only if the trace form $\operatorname{Tr}_{K}^{A}$ of the $K$-algebra $A:=K[X]/\langle f\rangle$ is positive definite.

For a partial generalization (see Theorem 4.10 below) of Theorem 4.5 and applications one can also consider the following more general trace forms :

4.9

Generalized trace forms *  Let

                       ${\rm Sym}_{K}(V,K):=\{\Phi\in{\rm Mult}_{K}\,(V,K)\mid\Phi\ \hbox{is symmetric}\,\}$

be the $K$-vector space of all symmetric bilinear forms on $V$ and consider a $K$-linear embedding

          $\,E_{A|K}:=\mathrm{Hom}_{K}(A,K)\longrightarrow{\rm Sym}_{K}(V,K)$, $\,\alpha\longmapsto\Phi_{\alpha}:A\times A\to K\;,\,(f,g)\mapsto\alpha(fg)\,$.

The elements of the image of this map are called generalized trace forms on $A$. The $A$-module $E_{A\,|K}$ ( with the scalar multiplication $(g\alpha)(f):=\alpha(fg)$ for $\alpha\in E$, $g$, $f\in A$) is called the dualizing module of $A$. Therefore :

4.9***.1***  $\Phi_{\alpha}(f,g)=(g\alpha)(f)=(f\alpha)(g)$ and the degeneration space $A^{{}^{{}_{\bot_{\alpha}}}}$ of $\,\Phi_{\alpha}$ is the largest ideal of $A$ contained in $\operatorname{Ker}\,\alpha\,$.

4.9***.2***  Let $\overline{\alpha}:A/A^{{}^{{}_{\bot_{\alpha}}}}\to K$ be the linear form on $\overline{A}:=A/A^{{}^{{}_{\bot_{\alpha}}}}$ induced by $\alpha$, then $\operatorname{rank}\Phi_{\alpha}=\operatorname{rank}\Phi_{\,\overline{\alpha}}$ and the induced bilinear form $\Phi_{\,\overline{\alpha}}$ is non-degenerate on $\overline{A}$.

4.9***.3***  Moreover, if $K$ is an ordered field, then :

                           $\operatorname{type}\Phi_{\alpha}=\operatorname{type}\Phi_{\,\overline{\alpha}}\,$ and $\,\operatorname{sign}\Phi_{\alpha}=\operatorname{sign}\Phi_{\,\overline{\alpha}}$.*

For example, for a fixed $h\in A$, the symmetric bilinear from $\Phi_{h}:A\times A\rightarrow K$, $(f,f^{\prime})\mapsto\operatorname{Tr}_{K}^{A}(hff^{\prime})$ is the generalized trace from on $A$ with respect to the $K$-linear form $\lambda_{\,h}:A\rightarrow A$, $g\mapsto hg$.

We shall use these particular generalized trace forms on $A$ and the following partial generalization of the Theorem 4.5  in the proof of Theorem 5.5.

4.10 Theorem

(cf. [3, Theorem 3.4])*  Let $\alpha$ be a $K$-linear form on a finite algebra $A$ over a real closed field $K$. If $\,\operatorname{sign}\Phi_{\alpha}\neq 0$, then $A$ has a $K$-rational point, i. e.  $K\operatorname{\!-Spec}A\neq\emptyset$.*

5. Counting rational points of 0-dimensional affine algebraic sets

In this section we will apply results from Section 4 on trace forms to count the rational points of finite affine algebraic sets over real closed fields. Our method is a modern version of old results of Hermite and Sylvester who had used signatures of quadratic forms to count real zeros of polynomials in one variable, see [7], [8] and [19]. We use elementary commutative algebra to treat multivariate versions of these problems.

5.1

Notation, Assumptions and Consequences*  Throughout this section, we use the following notation and assumptions and their consequences :*

*Let $K$ be a real closed field with notation as in 3.1. For an ideal $\mathfrak{A}\subseteq K[X_{1},\ldots,X_{n}]$ in the polynomial ring $K[X_{1},\ldots,X_{n}]$ over $K$, let

                           ${\rm V}_{K}(\mathfrak{A}):=\{a\in K^{n}\mid F(a)=0\ \hbox{ for all }\ F\in\mathfrak{A}\}$  and

                           $\,{\rm V}_{\mathds{K}}(\mathfrak{A}):=\{a\in\mathds{K}^{n}\mid F(a)=0\ \hbox{ for all }\ F\in\mathfrak{A}\}$.

be the affine algebraic set in $K^{n}$ and in $\mathds{K}^{n}$ defined by $\mathfrak{A}$, respectively.*

Polynomials in $K[X_{1},\ldots,X_{n}]$ are denoted by capital letters $F$, $G$, $H$, $\ldots$ and their images in the residue class $K$-algebra $A:=K[X_{1},\ldots,X_{n}]/\mathfrak{A}$ are denoted by small letters $f$, $g$, $h$, $\ldots$ .

*Every element $f\in A$ defines a (regular or polynomial) function on ${\rm V}_{K}(\mathfrak{A})$, namely $\,f:{\rm V}_{K}(\mathfrak{A})\longrightarrow K$, $a\longmapsto f(a)$. Further, if $f$, $g\in A$, then, clearly :

         $f=g$ on ${\rm V}_{K}(\mathfrak{A})\iff f=g$ in $A\iff\,F\equiv G~{}{\rm(\,mod\,}\mathfrak{A}\,)$, i. e. $F-G\in\mathfrak{A}$.*

We assume that the residue class $K$-algebra $\,A:=K[X_{1},\ldots,X_{n}]/\mathfrak{A}$ is finite dimensional $K$-vector space, or equivalently, the affine algebraic set ${\rm V}_{K}(\mathfrak{A})\subseteq K^{n}$ is a finite set. These assumptions are equivalent with the conditions : the $\mathds{K}$-algebra $\mathds{K}\otimes_{K}A=A_{\mathds{K}}=\mathds{K}[X_{1},\ldots,X_{n}]/\langle\mathfrak{A}\rangle$ is finite dimensional over $\mathds{K}$, or equivalently, the affine algebraic subset ${\rm V}_{\mathds{K}}(\mathfrak{A})\subseteq\mathds{K}^{n}$ is a finite set.

Further, since $\mathfrak{A}\subseteq K[X_{1},\ldots,X_{n}]$, it follows that if ${\bf a}\!\in\!{\rm V}_{\mathds{K}}(\mathfrak{A})$, then its conjugate $\overline{{\bf a}}\!\in\!{\rm V}_{\mathds{K}}(\mathfrak{A})$, too. Therefore, since ${\rm V}_{K}(\mathfrak{A})\subseteq{\rm V}_{\mathds{K}}(\mathfrak{A})$, renumbering we assume that :

**5.1.a ***  ${\rm V}_{K}(\mathfrak{A})\!=\!\{{\bf a}_{1},\ldots,{\bf a}_{r}\}\!\subseteq\!{\rm V}_{\mathds{K}}(\mathfrak{A})\!=\!\{{\bf a}_{1},\ldots,{\bf a}_{r}\,,\,{\bf a}_{r+1}\,,\,\overline{{\bf a}}_{r+1}\,,\,\ldots\,,\,{\bf a}_{r+s}\,,\,\overline{{\bf a}}_{r+s}\}$,

where $\,r:=\#{\rm V}_{K}(\mathfrak{A})$, $r+s=\#\operatorname{Spm}A$   and   $m:=r+2s=\operatorname{Dim}_{K}A=\operatorname{Dim}_{\mathds{K}}A_{\mathds{K}}=\#{\rm V}_{\mathds{K}}(\mathfrak{A})$.

Furthermore, since $K$ is a real closed field, ${\rm Char}\,K=0$, in particular, $K$ is infinite and hence by a linear change of coordinates (over $K$) (for instance, $Y_{i}=X_{i}$ for all $i=1,\ldots,n-1$ and $Y_{n}=X_{n}+\sum_{\,i=1}^{n-1}\,X_{i}\,t^{i}$ for suitable $t\in K$ avoiding finitely many), we may assume that ${\rm V}_{\mathds{K}}(\mathfrak{A})$ is in general $X_{n}$-position, or the ideal $\mathfrak{A}$ in general $X_{n}$-position  (The intention is to separate all zeros in an algebraic closure of $K$ by their last coordinate),  i. e. :

***5.1.b ***  The $n$-th coordinates $a_{i\,n}$ of the points ${\bf a}_{i}\!=\!(a_{i1},\ldots,a_{i\,n})\in\mathds{K}^{n}\!\!$, $i\!=\!1,\ldots,m$ are all distinct.

Note that ${\rm V}_{K}(\mathfrak{A})\!=\!{\rm V}_{\mathds{K}}(\mathfrak{A})\cap K^{n}$ is the set of $K$-rational points of {\rm V}_{\mathds{K}}(\mathfrak{A})\!\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}\!K\operatorname{\!-Spec}A_{\mathds{K}}=\operatorname{Spm}A_{\mathds{K}}=\mathrm{Spec}A_{\mathds{K}} (the first equality follows from Hilbert’s Nullstellensatz, see [12] or [6, Theorem 2.10, HNS 3] ) and {\rm V}_{K}(\mathfrak{A})\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}K\operatorname{\!-Spec}A\subseteq\operatorname{Spm}A\!=\!\mathrm{Spec}A, see 4.1 (b). Further, since $A$ and $A_{\mathds{K}}$ are reduced, the local components (see 4.1 (c)) of $A$ corresponding to the $K$-rational points ${\bf a}_{i}\!\in\!{\rm V}_{K}(\mathfrak{A})$, $i=1,\ldots,r$, are isomorphic to $K$ and corresponding to $\mathscr{M}\in\operatorname{Spm}A\!\smallsetminus\!K\operatorname{\!-Spec}A$ are isomorphic to $\mathds{K}$, but local components of $A_{\mathds{K}}$ corresponding to all the points ${\bf a}\in{\rm V}_{\mathds{K}}(\mathfrak{A})$ are all isomorphic to $\mathds{K}$. Therefore the explicit structures of the $K$-algebra $A$ and the $\mathds{K}$-algebra $A_{\mathds{K}}$ are determined by the algebra isomorphisms which are defined by the substitutions :

***5.1.c ***     \,A\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}\enskip K^{r}\times\mathds{K}^{s}\,, $h\mapsto\left(h\,(\,{\rm mod}\,\mathscr{M}\,)\right)_{\mathscr{M}\in\operatorname{Spm}A}$, where $r$, $s$ as in 5.1.a   and  

            \,A_{\mathds{K}}\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}\enskip\mathds{K}^{m}, $f\mapsto\left(f({\bf a})\right)_{{\bf a}\in{\rm V}_{\mathds{K}}(\mathfrak{A})}\,$, where $m:=r+2s$.

Note that $m=\operatorname{Dim}_{K}A=\operatorname{Dim}_{\mathds{K}}A_{\mathds{K}}=\#{\rm V}_{\mathds{K}}(\mathfrak{A})$.

Furthermore, the following eigenvector theorem (see [4, Ch. 2, §4, Theorem 4.5] which follows directly from 5.1.b :

***5.1.d *** For every $h\!\in\!A$, the eigenvalues of the $K$-linear map $\lambda_{h}:A\to A$, $f\mapsto hf$ are the values $h({\bf a}_{1}),\ldots,h({\bf a}_{r})$, $\,h({\bf a}_{r+1}),h(\overline{{\bf a}}_{r+1})\ldots,h({\bf a}_{r+s}),h(\overline{{\bf a}}_{r+s})$ of the function $h:{\rm V}_{\mathds{K}}(\mathfrak{A})\rightarrow\mathds{K}$.

For more accessible determination of the signature of the trace form $\,\operatorname{Tr}^{A}_{K}$, we need a nice basis of $A$ over $K$. The following crucial key observation so-called Shape Lemma (see [4], [5] and [11]) guarantees a distinguished generating set for a radical ideal $\mathfrak{A}$ in $K[X_{1},\ldots,X_{n}]$. We give a proof of the Shape Lemma by using the natural action of the Galois group $\operatorname{Gal}(\overline{K}|K)$ on ${\rm V}_{\overline{K}}(\mathfrak{A})$.*

5.2 Lemma

( Shape Lemma )*  Let $K$ be an infinite perfect field and $\mathfrak{A}\subseteq K[X_{1},\ldots,X_{n}]$ be a radical ideal and let $A:=K[X_{1},\ldots,X_{n}]/\mathfrak{A}$ be a finite dimensional $K$-vector space. With further notation and assumptions as in 5.1. There exist polynomials $g_{1},\ldots,g_{n-1},\,g_{n}\in K[T]\,($where $T$ is indeterminate over $K)$ with $g_{n}\neq 0$ square free of degree $m$, such that $\,\mathfrak{A}$ is generated by $X_{1}-g_{1}(X_{n})\,,\ldots,\,X_{n-1}-g_{n-1}(X_{n})$, $g_{n}(X_{n})\,$. In particular, $\,\mathbcal{x}=\{1,x_{n},\ldots,x_{n}^{m-1}\}$ is a $K$-basis of $A$, where $x_{n}$ is the image of $X_{n}$ in $A$.*

Proof  Let $\overline{K}$ be an algebraic closure of $K$. Since $K$ is perfect, the field extension $\overline{K}|K$ is a Galois extension. Let $\operatorname{Gal}(\overline{K}|K)$ be its Galois group.

Let ${\rm V}_{\overline{K}}(\mathfrak{A})\!:=\!\{a\in\overline{K}^{n}\!\mid F(a)\!=0\!\hbox{ for all }F\!\in\!\mathfrak{A}\}$. Then ${\rm V}_{\overline{K}}(\mathfrak{A})$ is a finite set by assumption on $\mathfrak{A}$ and the projection map $q:{\rm V}_{\overline{K}}(\mathfrak{A})\rightarrow\overline{K}$, $(a_{1},\ldots,a_{n})\mapsto a_{n}\,$ is injective (by assumption (see 5.1.b)), (i. e. $q\,$ separates points in ${\rm V}_{\overline{K}}(\mathfrak{A})$).

The Galois group ${\rm Gal}\,(\overline{K}|K)$ operates on ${\rm V}_{\overline{K}}(\mathfrak{A})$ with the natural operation :

        $\,{\rm Gal}\,(\overline{K}|K)\times{\rm V}_{\overline{K}}(\mathfrak{A})\longrightarrow{\rm V}_{\overline{K}}(\mathfrak{A})$, $(\sigma,(a_{1},\ldots,a_{n}))\longmapsto(\sigma(a_{1}),\ldots,\sigma(a_{n}))$.

Obviously, the image $q({\rm V}_{\overline{K}}(\mathfrak{A}))\!=\!W_{1}\uplus\cdots\uplus W_{\ell}$ is the union of orbits of this operation and each orbit $W_{k}\!=\!{\rm V}_{\overline{K}}(\pi_{k})$ is the zero set of the irreducible polynomial $\pi_{k}\!\in\!K[T]$, $k\!=\!1,\ldots,\ell$,  see [9]  or  [17, Ch. XI, §93, 93.2]. Therefore, since $K$ is perfect, the polynomial $g_{n}:=\pi_{1}\cdots\pi_{\ell}\in K[T]$ is square free and $q({\rm V}_{\overline{K}}(\mathfrak{A}))\!=\!{\rm V}_{\overline{K}}(g_{n})$, $\deg\,g_{n}\!=\!\#{\rm V}_{\overline{K}}(\mathscr{M})\!=\!m$.

5.2.a  For all $\,a_{n}\in q({\rm V}_{\overline{K}}(\mathfrak{A}))$, there exist polynomials $\,g_{1},\ldots,g_{n-1}\in K[T]$ with $\deg g_{i}<\deg g_{n}=m$ such that $(g_{1}(a_{n}),\ldots,g_{n-1}(a_{n}),a_{n})$ is the unique point lying over $a_{n}$.

To prove 5.2.a, let $a_{n}\in q({\rm V}_{\overline{K}}(\mathfrak{A}))$ and $(a_{1},\ldots,a_{n-1},\,a_{n})$ be the unique point lying over $a_{n}$. We may assume that $a_{n}\in W_{1}=\{\sigma_{j}(a_{n})\mid j=1,\ldots,d_{1},\,\sigma_{1}=\operatorname{id}_{\overline{K}}\}$ with $d_{1}=\#\,W_{1}$. Let $W^{\prime}_{i}$ denote the orbit of $a_{i}$. Then, since $q$ is injective, $\#\,W^{\prime}_{i}\leq\#\,W_{1}=d_{1}$. Moreover, for all $i=1,\ldots,n-1$, $W^{\prime}_{i}=\{\sigma_{j}(a_{i})\mid j=1,\ldots,d_{1},\,\sigma_{1}=\operatorname{id}_{\overline{K}}\}$, but all $\sigma_{j}(a_{i})$, $j=1,\ldots,d_{1}$, may not be distinct.

Now, since $\sigma_{j}(a_{n})$, $j\!=\!1,\ldots,d_{1}$, are distinct elements in $\overline{K}$, by Lagrange’s Interpolation Formula 888 Although named after Lagrange, J. L. (1736 - 1813) who published it in 1795, the method was first discovered in 1779 by Waring, E. (1734 - 1798). It is also an easy consequence of a formula of Euler, L. (1707 - 1783) published in 1783.    Lagrange’s Interpolation Formula : Let $K$ be a field and let $x_{1},\ldots,x_{n}\in K$ be distinct elements. Then for arbitrary elements $y_{1},\ldots,y_{n}\in K$, there exists a polynomial $g\in K[X]$ of degree $\deg\,g<n$ such that $g(x_{i})=y_{i}\,$ for every $i=1,\ldots,n$. For a proof consider the polynomial $\,\displaystyle g:=\sum_{i=1}^{n}\,\,\frac{y_{i}}{z_{i}}\,\prod_{j\neq i}\,\,(X-x_{j})$, where $\,\displaystyle z_{i}:=\prod_{j\neq i}\,\,(x_{i}-x_{j})$. , for each $i\!=\!1,\ldots,n-1$, there exists a polynomial $g_{i}\!\in\!\overline{K}[X]$, $\deg\,g_{i}\!<\!d_{1}\!<\!\deg\,g_{n}$, such that $g_{i}(\sigma_{j}(a_{n}))\!=\!\sigma_{j}(a_{i})$ for all $j\!=\!1,\ldots,d_{1}\!$. Moreover, $g_{1},\ldots,g_{n-1}\!\in\!K[X]$.

Finally we claim the equality $\mathfrak{A}^{\prime}:=\langle X_{1}-g_{1}(X_{n}),\ldots,X_{n-1}-g_{n-1}(X_{n}),\,g_{n}(X_{n})\rangle=\mathfrak{A}$. To prove this first note that the substitution homomorphism $K[X_{1},\ldots,X_{n-1},X_{n}]\rightarrow K[X_{n}]$, $X_{i}\mapsto X_{i}-g_{i}(X_{n})$, $i=1,\ldots,n-1$ and $X_{n}\mapsto g_{n}(X_{n})$, induces a $K$-algebra isomorphism K[X_{1},\ldots,X_{n}]/\mathfrak{A}^{\prime}\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}K[X_{n}]/\langle g\rangle and $K[X_{1},\ldots,X_{n}]/\mathfrak{A}^{\prime}$ is reduced, since $g_{n}$ is separable over $K$. Therefore $\mathfrak{A}^{\prime}$ is a radical ideal. Further, from 5.2.a it follows that ${\rm V}_{\overline{K}}(\mathfrak{A}^{\prime})={\rm V}_{\overline{K}}(\mathfrak{A})$. Now, use Hilbert’s Nullstellensatz (see [2], [12] or [15, Theorem 2.10 , HNS 2]) to conclude the equality $\mathfrak{A}^{\prime}=\mathfrak{A}$.∎

5.3 Remark

*The Shape Lemma 5.2 appeared first time in [5] which may be regarded as a natural generalization of the Primitive Element Theorem. Further, it gives a very useful presentation of the radical ideal $\mathfrak{A}$ which allows to find the solution space ${\rm V}_{\overline{K}}(\mathfrak{A})$ immediately, namely :

                  $\,{\rm V}_{\overline{K}}(\mathfrak{A})=\{(g_{1}(a),\ldots,g_{n-1}(a),a)\in\overline{K}^{n}\mid g_{n}(a)=0\,\}$.

In other words the last coordinates are zeros of $g_{n}$ and for a fixed last coordinate $a_{n}$, all the other coordinates are determined by evaluation of polynomials $g_{n-1},\ldots,g_{1}$ at $a_{n}$ :   $\,g_{n}(a_{n})=0$, $\,a_{n-1}=g_{n-1}(a_{n}),\,\ldots\,,a_{1}=g_{1}(a_{n})$. This simple shape of the solution space $\,{\rm V}_{\overline{K}}(\mathfrak{A})$ is quite convenient to work with. The primary decomposition of $\mathfrak{A}$ is given by the prime factorization of the polynomial $g_{n}$. Under the conditions on the polynomials $g_{1},\ldots,g_{n-1}$, $g_{n}\in K[X]$ as in the proof of the Shape Lemma 5.2, one can easily verify that $\,X_{1}-g_{1}(X_{n}),\ldots,X_{n-1}-g_{n-1}(X_{n}),\,g_{n}(X_{n})\,$ form a reduced (= minimal) Gröbner basis of the radical ideal $\,\mathfrak{A}\,$ relative to the lexicographic order $X_{1}>X_{2}>\cdots>X_{n}$. For a different proof of the Shape Lemma 5.2 see [11, Theorem 3.7.25] and a detailed recipe for solving systems of polynomial equations efficiently using the Shape Lemma 5.2 is also given in [11, Theorem 3.7.26]. The Shape Lemma 5.2 also appeared in [4, Ex. 16, § 4, Ch. 2].*

5.4

Consequence and identifcation*  Let $K$ be a real closed field, $\mathds{K}:=\mathds{C}_{K}=K[\,{\rm i}\,]$, ${\rm i}^{2}=-1$, the algebraic closure of $K$ (see 3.1) and let $\mathfrak{A}\subseteq K[X_{1},\ldots,X_{n}]$ a radical ideal. Suppose that $A:=K[X_{1},\ldots,X_{n}]/\mathfrak{A}$ is a finite dimensional $K$-vector space.*

Let $g_{1},\ldots,g_{n-1},g:=g_{n}(X)\in K[X]$ are the polynomials as in the statement of the Shape Lemma 5.2 and let \varphi:A\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}K[X]/\langle g\rangle be the $K$-algebra isomorphism as in the proof of the Shape Lemma 5.2. Then, since $g$ is square-free and $K$ is a real closed field (see Footnote 2), $g=(X-a_{1})\cdots(X-a_{r})\pi_{1}\cdots\pi_{s}\,$, $a_{i}\in K$, $i=1,\ldots r\,$ and $\pi_{j}=(X-z_{j})(X-\overline{z}_{j})\in K[X]$, $z_{j}\!\in\!\mathds{K}\!\smallsetminus K$, $j\!=\!1,\ldots,s$, where $r$, $s$ and $m\!=\!r\!+\!2s$ as in 5.1.a*, since $\varphi$ is a $K$-algebra isomorphism.*

We use the above $K$-algebra isomorphism $\varphi$ to identify $\mathfrak{A}$ and $A$ with $\langle g\rangle$ and $K[X]/\langle g\rangle$, respectively. With this $\mathbcal{x}:=\{1,x,\ldots,x^{\,m-1}\}$ is a $K$-basis of $A$, where $x$ is the image of $X$ in $A$ and ${\rm V}_{K}(\mathfrak{A})={\rm V}_{K}(g)=\{a_{1},\ldots,a_{r}\}\subseteq{\rm V}_{\mathds{K}}(\mathfrak{A})={\rm V}_{\mathds{K}}(g)=\{a_{1},\ldots,a_{r},z_{1},\overline{z}_{1},\ldots,z_{s},\overline{z}_{s}\}$, $r+2s=m$.

Further, for $H\in K[X_{1},\ldots,X_{n}]$, $H\neq 0$, we put $h(X):=H(g_{1}(X),\ldots,g_{n-1}(X),X)\in K[X]$. Then using the above identifications, we have $h(x)\in A$, and the values $H({\bf a}_{i})\in K$, $i=1,\ldots,r$, and $H({\bf a}_{r+j})$, $H(\overline{\bf a}_{r+j})\in\mathds{K}$, $j=1,\ldots,s$ are identified with the values $h(a_{i})\in K$, $i=1,\ldots,r$, and $h(z_{j})$, $h(\overline{z}_{j})\in\mathds{K}$, $j=1,\ldots,s$, respectively.

5.5 Theorem

With the notation as in 5.1, in 5.4, let $H\in K[X_{1},\ldots,X_{n}]$, $H\neq 0$, $h$ be the image of $H$ in $A$ and let $\,\Phi_{h}:A\times A\to K$, $(f,f^{\prime})\mapsto\operatorname{Tr}_{K}^{A}(hff^{\prime})$, be the generalized trace form associated with $h\in A$. Then $:$

• (a)

*The Gram’s matrix $\mathscr{G}_{\Phi_{h}}(\mathbcal{x})$ of $\,\Phi_{h}$ with respect to the $K$-basis $\mathbcal{x}$ is a symmetric matrix in ${\rm M}_{m}(K)$. Moreover, $\,\mathscr{G}_{\Phi_{h}}(\mathbcal{x})\!=\!\mathscr{V}\,\mathscr{D}\,{}^{\rm t}\mathscr{V}$, where $\mathscr{V}\in\operatorname{GL}_{m}(\mathds{K})$ is the Vandermonde’s matrix *999 Vandermonde’s matrix  For elements $a_{1},\ldots,a_{m}$ is a field $K$, the matrix $\mathscr{V}(a_{1},\ldots,a_{m}):=(a_{i}^{\,j})_{\genfrac{}{}{0.0pt}{5}{1\leq i\leq m}{0\leq j\leq m-1}}\in\operatorname{M}_{m}(K)$ is called the Vanderminde’s matrix of the elements $a_{1},\ldots,a_{m}$. The elements $a_{1},\ldots,a_{m}$ are pairwise distinct if and only if $\mathscr{V}(a_{1},\ldots,a_{m})\in\operatorname{GL}_{m}(K)$.

• of the elements $a_{1},\ldots,a_{r},z_{1},\ldots,z_{s},\overline{z}_{1},\ldots,\overline{z}_{s}\in\mathds{K}$  and  $\mathscr{D}\in\operatorname{M}_{m}(\mathds{K})$ is the diagonal matrix with diagonal entries $\,h(a_{1}),\ldots,h(a_{r}),h(z_{1}),\ldots,h(z_{s}),h(\overline{z}_{1}),\ldots,h(\overline{z}_{s})$.*

• (b)

Let $\,p_{H}:=\#\,\{{\bf a}\in{\rm V}_{K}(\mathfrak{A})\mid H({\bf a})>0\,\}\,$ and $\,q_{H}:=\#\,\{{\bf a}\in{\rm V}_{K}(\mathfrak{A})\mid H({\bf a})<0\,\}\,$. Then $\,\operatorname{type}\,\Phi_{h}\!=\!(p_{H}+s\,,\,q_{H}+s)$, where $s\!=\!\#\left({\rm V}_{\mathds{K}}(\mathfrak{A})\!\smallsetminus\!{\rm V}_{K}(\mathfrak{A})\right)$ and $\operatorname{rank}\Phi_{h}\!=\,\#\{{\bf a}\in{\rm V}_{\mathds{K}}(\mathfrak{A})\mid H({\bf a})\neq 0\}$. In particular, $\,\operatorname{sign}\,\Phi_{h}=p_{H}-q_{H}\,$.

Proof  Recall from 5.1 that :

$\,{\rm V}_{K}(\mathfrak{A})=\{{\bf a}_{1},\ldots,{\bf a}_{r}\}\subseteq{\rm V}_{\mathds{K}}(\mathfrak{A})=\{{\bf a}_{1},\ldots,{\bf a}_{r}\,,\,{\bf a}_{r+1}\,,\,\overline{{\bf a}}_{r+1}\,,\,\ldots\,,\,{\bf a}_{r+s}\,,\,\overline{{\bf a}}_{r+s}\}$, where $r:=\#\,{\rm V}_{K}(\mathfrak{A})$, $r+s=\#\,\operatorname{Spm}\,A$ and $m=r+2s=\operatorname{Dim}_{K}A=\operatorname{Dim}_{\mathds{K}}A_{\mathds{K}}=\#\,{\rm V}_{\mathds{K}}(\mathfrak{A})$ and that ${\rm V}_{\mathds{K}}(\mathfrak{A})$ is in general $X_{n}$-position, see 5.1.a and 5.1.b.

(a)  From the indentifcations in  5.4, it follows that for $1\leq k\,,\,\ell\leq m-1$, the $(k,\ell)$-entry in the Gram’s matrix $\mathscr{G}_{\Phi_{h}}(1,x,\ldots,x^{m-1})$ is :

5.5**.1**     $\displaystyle\operatorname{Tr}_{K}^{A}(h(x)\,x^{k+\ell-2})=\sum_{z\in{\rm V}_{\mathds{K}}(g)}\!\!h(z)\,z^{\,k+\ell-2}\,$

                               $\displaystyle\,=\,\sum_{i=1}^{r}\!h(a_{i})\,a_{i}^{\,k+\ell-2}+\sum_{j=1}^{s}\!\left(h(z_{j})\,z_{j}^{\,k+\ell-2}+h(\overline{z}_{j})\,\overline{z}_{j}^{\,k+\ell-2}\right)$.

Now, by the Fundamental Theorem on Symmetric Polynomials (see [17, Theorem 54.13], the right hand side of 5.5.1 is a polynomial in the coefficients of $h(X)$ and $g(X)$ (with coefficients in $\mathds{Z}$) and hence belongs to $K$. Therefore $\,\mathscr{G}_{\Phi_{h}}(1,x,\ldots,x^{m-1})\,$ is a symmetric matrix in ${\rm M}_{m}(K)$. Furthermore, using the equation 5.5.1, the equality $\,\mathscr{G}_{\Phi_{h}}(1,x,\ldots,x^{m-1})=\mathscr{V}\,\mathscr{D}{}^{\rm t}\mathscr{V}$, where $\mathscr{V}$ and $\mathscr{D}$ are as in the statement of (a), can be easily verified.

(b)  The assertion about the rank follows from the equality $\operatorname{rank}\Phi_{h}=\operatorname{rank}\mathscr{G}_{\Phi}(\mathbcal{x})=\operatorname{rank}\mathscr{D}$, since $\mathscr{V}\in\operatorname{GL}_{m}(\mathds{K})$. Further, note that the local decomposition \,A\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}K^{r}\times\mathds{K}^{s} (see 5.1.c) yields the orthogonal decomposition (see 4.2 and 2.13)

                       \Phi_{h}\!=\!(\Phi_{h})_{1}^{K}\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}\cdots\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}(\Phi_{h})_{r}^{K}\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}(\Phi_{h})_{1}^{\mathds{K}}\mathbin{\hbox{\vrule width=0.0pt,height=5.83333pt,depth=0.83333pt\kern 0.44337pt\vrule height=2.62497pt,depth=-2.04169pt,width=6.59563pt\kern 0.77782pt\hbox to0.0pt{\hss\hbox{\vrule width=0.58336pt,depth=-2.33328pt,height=5.83333pt\kern 3.84996pt}}\hbox to0.0pt{\hss\hbox{\odot}}}}\cdots(\Phi_{h})_{s}^{\mathds{K}},

where $(\Phi_{h})_{i}^{K}\!=\!\Phi_{h}|K$, is the restrictions of $\Phi_{h}$ to the real component at $a_{i}\!\in\!K$ with Gram’s matrix $\mathscr{G}_{(\Phi_{h})_{i}^{K}}(1)\!=\!(h(a_{i}))\in{\rm M}_{1}(K)$, $i\!=\!1,\ldots,r$) and $(\Phi_{h})^{\mathds{K}}_{j}\!=\!\Phi_{h}|\mathds{K}$, is the restrictions of $\Phi_{h}$ to the non-real component \,K[X]/\langle\pi_{j}\rangle\stackrel{{\scriptstyle\raise 1.0pt\hbox{\mathchoice{\vbox to0.0pt{\hbox{$\displaystyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\textstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptstyle{\sim}$}\vss}}{\vbox to0.0pt{\hbox{$\scriptscriptstyle{\sim}$}\vss}}}}}{{\longrightarrow}}\mathds{K}\, at $\mathscr{M}_{j}\!=\!\langle\pi_{j}\rangle\in\operatorname{Spm}\,A\!\smallsetminus\!K\operatorname{\!-Spec}A$, $j\!=\!1,\ldots,s$. Furthermore, clearly, $\operatorname{type}\,(\Phi_{h})_{i}^{K}\!=\!\operatorname{sign}(\Phi_{h})_{i}^{K}\!=\!\operatorname{sign}h(a_{i})\!=\!\operatorname{sign}\,H({\bf a}_{i})$ for all $i\!=\!1,\ldots,r$ and by Example 3.10 (since $\pi_{j}\!=\!(X\!-\!z_{j})(X\!-\!\overline{z}_{j})$, $z_{j}\in\mathds{K}\!\smallsetminus\!K$), we have $\,\operatorname{type}\,(\Phi_{h})_{j}^{\mathds{K}}\!=\!(1,1)$ for all $j\!=\!1,\ldots,s$. Therefore, by Corollary 4.4 :

                 $\operatorname{type}\,\Phi_{h}\!=\!\sum_{i=1}^{r}\,\operatorname{type}\,(\Phi_{h})_{i}^{K}\!+\!\sum_{j=r+1}^{r+s}\,\operatorname{type}\,(\Phi_{h})_{j}^{\mathds{K}}\!=\!(p_{H}+s\,,\,q_{H}+s)$

and hence $\,\operatorname{sign}\,\Phi_{h}\!=\!p_{H}\!-\!q_{H}$. ∎

5.6 Corollary

(Hermite)* Let $K$ be an arbitrary real closed field and let $\,g=b_{0}+b_{1}X+\cdots+b_{m-1}X^{m-1}+X^{m}\in K[X]$, $\,A:=K[X]/\langle g\rangle$. Then the $\,\operatorname{type}\,\operatorname{Tr}_{K}^{A}=(r+s\,,\,s)$, where $\,\operatorname{Tr}_{K}^{A}:A\times A\to K$, $(f,f^{\prime})\mapsto\operatorname{Tr}_{K}(ff^{\prime})$ is the trace form on $A$, $\,r=\#{\rm V}_{K}(g)$ is the number of zeros of $g$ in $K$ and $\,s\,$ is the half of the number of zeros of $g$ in the algebraic closure $\mathds{K}$ of $K$ which are not in $K$. In particular, $\operatorname{sign}\,\operatorname{Tr}_{K}^{A}=r=\#\,{\rm V}_{K}(g)$.*

Proof  Using the notations as in the Theorem 5.5, note that $\operatorname{Tr}_{K}^{A}\!=\!\Phi_{1}$, where $1\!\in\!K[X]$ denote the constant polynomial. Therefore $p_{1}\!=\!r\!=\!\#{\rm V}_{K}(g)$, $q_{1}\!=\!0$ and $\operatorname{sign}\,\operatorname{Tr}_{K}^{A}\!=\!p_{1}-q_{1}\!=\!r\!=\!\#{\rm V}_{K}(g)$ by by 5.5 (c). Of course, the assertion also follows directly from Theorem 4.5. ∎

With the notation and assumptions as in 5.1, our main goal is to relate the cardinality $\#\,{\rm V}_{K}(\mathfrak{A})\,$ with the signatures of the generalized trace form on the finite $K$-algebra $A$.

5.7

Notation*  With the notation and assumptions as in 5.1 and as 5.4. Further, let $H\in K[X_{1},.....,X_{n}]$, $H\neq 0$ and ${\rm V}_{K}(H):=\{a\in K^{n}\mid H(a)=0\}$ be the hypersurface (an $(n-1)$-dimensional affine algebraic set in $K^{n}$) defined by $H$. Then the complement of ${\rm V}_{K}(H)$ in $K^{n}$ is the union of line-connected subsets (in the strong topology on $K^{n}$ (see Footnote 5) on which $H$ takes either all positive values or all negative values, i. e. $K^{n}\!\smallsetminus\!{\rm V}_{K}(H)\!=\!H^{+}\uplus H^{-}\!$, where $H^{+}\!:=\!\{a\in K^{n}\mid H(a)>0\}$, and $H^{-}\!:=\!\{a\in K^{n}\mid H(a)<0\}$.*

*Further, since ${\rm V}_{K}(\mathfrak{A})=\left({\rm V}_{K}(\mathfrak{A})\cap H^{+}\right)\biguplus\left({\rm V}_{K}(\mathfrak{A})\cap H^{-}\right)\biguplus\left({\rm V}_{K}(\langle\mathfrak{A},H\rangle)\right)$, we have :

5.7***.a***    $\,\#\,{\rm V}_{K}(\mathfrak{A})=\#\,\left({\rm V}_{K}(\mathfrak{A})\cap H^{+}\right)+\,\#\,\left({\rm V}_{K}(\mathfrak{A})\cap H^{-}\right)+\,\#\,\left({\rm V}_{K}(\langle\mathfrak{A},H\rangle)\right)\,$,

and hence to compute $\#\,{\rm V}_{K}(\mathfrak{A})\,$, we can use arbitrary polynomial $H\in K[X_{1},\ldots,X_{n}]$ and compute the cardinalities  $\#\,{\rm V}_{K}(\mathfrak{A})\cap H^{+}$,  $\#\,{\rm V}_{K}(\mathfrak{A})\cap H^{-}$ and  $\#\,{\rm V}_{K}(\langle\mathfrak{A},H\rangle)$.*

More precisely, we have :

5.8 Theorem

With the notation and assumptions as in 5.1 and 5.7. For $H\in K[X_{1},.....,X_{n}]$, $H\neq 0$, let $p_{H}:=\#\,{\rm V}_{K}(\mathfrak{A})\cap H^{+}$ and $q_{H}:=\#\,{\rm V}_{K}(\mathfrak{A})\cap H^{-}$. Further, let $h$ denote the image of $H$ in $A=K[X_{1},.....,X_{n}]/\mathfrak{A}$ and $\Phi_{h}:A\times A\to K$, $(f,g)\mapsto\operatorname{Tr}_{K}^{A}(hfg)$ $($resp. $\Phi_{h^{2}}:A\times A\to K$, $(f,g)\mapsto\operatorname{Tr}_{K}^{A}(h^{2}fg))$ be the generalized trace forms defined by $h\,($resp. by $h^{2})$ on $A$. Then :

• (a)

(Pederson-Roy-Szpirglas  [16, Theorem 2.1])*  *

               $\,\operatorname{sign}\,\Phi_{h}=p_{H}-q_{H}\,$  and  $\,\operatorname{rank}\,\Phi_{h}=\#\,\left({\rm V}_{\mathds{K}}(\mathfrak{A})\!\smallsetminus\!{\rm V}_{\mathds{K}}(H)\right)$.*

• (b)

$\,\operatorname{sign}\,\Phi_{h^{2}}=p_{H}+q_{H}\,$*  and  $\,\operatorname{rank}\,\Phi_{h^{2}}=\#\,\left({\rm V}_{\mathds{K}}(\mathfrak{A})\!\smallsetminus\!{\rm V}_{\mathds{K}}(H)\right)$.*

• (c)

*Let $\mathfrak{B}:=\langle\mathfrak{A},H\rangle$ be the ideal *$($in $K[X_{1},\ldots,X_{n}]\,)$ generated by $\mathfrak{A}$ and $H$. Then the $K$-algebra $B:=K[X_{1},\ldots,X_{n}]/\mathfrak{B}$ is finite over $K$ and $\,\operatorname{sign}\,\operatorname{Tr}^{B}_{K}=\#\,{\rm V}_{K}(\mathfrak{B})$.

• (d)

The three signatures $\,\operatorname{sign}\,\Phi_{h}$, $\,\operatorname{sign}\,\Phi_{h^{2}}$ and $\,\operatorname{sign}\,\operatorname{Tr}_{K}^{B}$ uniquely determine the natural numbers $p_{H}$, $q_{H}\,$  and   $\,\#\,{\rm V}_{K}(\mathfrak{B})={\rm V}_{K}(\mathfrak{A})\cap{\rm V}_{K}(H)$. In particular, they determine the cardinality $\,\#\,{\rm V}_{K}(\mathfrak{A})=p_{h}+q_{h}+\#\,{\rm V}_{K}(\mathfrak{B})$.

Proof  (a) : Proved in Theorem 5.5 (b).

(b) : Since $H^{2}(a)\!\!=\!H(a)\,H(a)\!>\!0$ for every $a\!\in\!H^{+}\!\cup H^{-}\!$ and ${\rm V}_{K}(H^{2})\!=\!{\rm V}_{K}(H)$, from Theorem 5.5 (b) it follows that :

              $\,\operatorname{sign}\,\Phi_{h^{2}}=p_{H}+q_{H}\,$   and $\,\operatorname{rank}\,\Phi_{h^{2}}\!\!=\!\#\,\left({\rm V}_{K}(\mathfrak{A})\!\smallsetminus\!{\rm V}_{K}(H)\right)$.

(c) : Since the $K$-algebra $B$ is a homomorphic image of the $K$-algebra $A$, $B$ is also finite over $K$. The equality $\,\operatorname{sign}\,\operatorname{Tr}^{B}_{K}=\#\,{\rm V}_{K}(\mathfrak{B})$ is immediate from Theorem 5.5 (a) ($H=1$) or Theorem 4.5.

(d) : Immediate from the formula 5.7.a for $\#{\rm V}_{K}(\mathfrak{A})$ in 5.7 and (a) and (b). ∎