Elimination ideals and Bezout relations

TL;DR

This paper investigates bounds on the degrees of elimination ideals and Bezout relations for polynomial ideals over infinite fields, focusing on the minimal degrees of polynomials depending on a subset of variables.

Contribution

It provides new bounds on the minimal degrees of polynomials in elimination ideals and Bezout relations, advancing understanding of polynomial ideal structure.

Findings

Established bounds on degrees of elimination polynomials.

Derived bounds for degrees of Bezout relations.

Enhanced understanding of polynomial ideal generators.

Abstract

Let $k$ be an infinite field and $I\subset k [x_1, \ldots ,x_n]$ be an ideal such that dim $V(I)=q$. Denote by $(f_1, \ldots, f_s)$ a set of generators of $I$. One can see that in the set $I\cap k [x_{1},...,x_{q+1}]$ there exist non-zero polynomials, depending only on these $q+1$ variables. We aim to bound the minimal degree of the polynomials of this type, and of a B\'ezout (i.e. membership) relation expressing such a polynomial as a combination of the $f_i$.

Equations

$$\pi:X\ni(x_{1},\dots,x_{m})\to\Big{(}\sum_{j=1}^{m}a_{1j}x_{j},\sum_{j=2}^{m}a_{2j}x_{j},\dots,\sum_{j=n}^{m}a_{1j}x_{j}\Big{)}\in\hbox{\tenmsy K}^{n}$$

$$\Phi :\text{tenmsy}{K}^n\times \text{tenmsy}K\ni (x,z)\to (F_1(x)z,\dots ,F_{n-q}(x)z,x)\in \text{tenmsy}{K}^{n-q}\times \text{tenmsy}{K}^n$$

$$\Psi =(\sum _{j=1}^{n-q}{\gamma }_{1j}{F}_{j}z+l_1(x),\dots ,\sum _{j=n-q}^{n-q}{\gamma }_{n-qj}{F}_{j}z+l_{n-q}(x),l_{n-q+1}(x),...,l_{n+1}(x)),$$

$$W({\Psi }_1,\dots ,{\Psi }_{n+1})=0\mathrm{and}\mathrm{deg}W(T_1^{d_1},T_2^{d_2},\dots ,T_k^{d_k},T_{k+1},\dots ,T_{n+1})\le d_s\prod _{j=1}^{n-q-1}{d}_j,$$

$$\pi :V(J)\ni x\mapsto (x_1,...,x_{q+1})\in {\text{tenmsy}K}^{q+1}.$$

$${\mathrm{deg}}_T{P}_z(T_1^{d_1},T_2^{d_2},\dots ,T_n^{d_n},T_{n+1},Y)\le d_s\prod _{j=1}^{n-q-1}{d}_j$$

$$\pi :V(J)\ni x\mapsto (x_1,...,x_{q+1})\in {\text{tenmsy}K}^{q+1}.$$

$$\pi :V(I)\ni x\mapsto (x_1,...,x_{q+1})\in {\text{tenmsy}K}^{q+1}$$

$$\pi :V(I)\ni x\mapsto (x_1,...,x_{q+1})\in {\text{tenmsy}K}^{q+1}.$$

$$X_i=x_i+t\sum _{j=i+1}^n{a}_{i,j}{x}_j;x_i=X_i+t\sum _{j=i+1}^n{b}_{i,j}(t)X_j,$$

$$b_0(X,t)=\sum _{j=1}^{n-q}{G}_j(X,t)\overline{F_j}(X,t),$$

$$b_0(X,t)=\sum _{j=1}^{n-q}(G_j(X,t)-\overline{H_j}(X,t))\overline{F_j}(X,t).$$

Full text

1. Introduction

Let $I\subset k[x_{1},...,x_{n}]$ be a non-zero ideal such that dim $V(I)=q\geq 0$. Using Hilbert Nullstellensatz we can easily see, that in the elimination ideal $I\cap k[x_{1},...,x_{q+1}]$ there exist non-zero polynomials. It is interesting to know the minimal degree of the polynomials in this ideal. Here, performing a generic change of coordinates, and continuing the approach presented in [1], we get a sharp estimate for the degree of such a minimal polynomial (and also for a corresponding generalized Bezout identity), in terms of the degrees of generators of the ideal $I$. Then, using a deformation arguments we solve the stated problem. We show that if $\deg f_{i}=d_{i}$ where $d_{1}\geq d_{2}...\geq d_{s},$ then there exist polynomials $g_{j}\in k[x_{1},\dots,x_{n}]$ and a non-zero polynomial $\phi(x)\in k[x_{1},...,x_{q+1}]$ such that

(a) $\deg g_{j}f_{j}\leq d_{s}\prod^{n-q-1}_{i=1}d_{i},$

(b) $\phi(x)=\sum^{s}_{j=1}g_{j}f_{j}.$

Note that our result works also in the case dim $V(I)=-1$ (i.e. in the case when $V(I)=\emptyset$) if we put $k[x_{0}]:=k$ (however our result in this case is a little bit worse than these in [1], [2]). Hence, from this point of view, we can treat our result as a generalization of the Effective Nullstellensatz.

2. Main Result

In this section we present a geometric construction and establish degree bounds, relying on generic changes of coordinates. Let us recall (see [1]) two important tools that we will use in the proof of the main theorem of this section.

Theorem 1**.**

(Perron Theorem) Let 𝕃 be a field and let $Q_{1},\dots,Q_{n+1}\in\hbox{\tenmsy L}[x_{1},\dots,x_{n}]$ be non-constant polynomials with $\deg Q_{i}=d_{i}$. If the mapping $Q=(Q_{1},\dots,Q_{n+1}):\hbox{\tenmsy L}^{n}\to\hbox{\tenmsy L}^{n+1}$ is generically finite, then there exists a non-zero polynomial $W(T_{1},\dots,T_{n+1})\in\hbox{\tenmsy L}[T_{1},\dots,T_{n+1}]$ such that

(a) $W(Q_{1},\dots,Q_{n+1})=0$,

(b) deg $W(T_{1}^{d_{1}},T_{2}^{d_{2}},\dots,T_{n+1}^{d_{n+1}})\leq\prod^{n+1}_{j=1}d_{j}.$

Lemma 2**.**

Let 𝕂 be an algebraic closed field and let $k\subset\hbox{\tenmsy K}$ be its infinite subfield. Let $X\subset\hbox{\tenmsy K}^{m}$ be an affine algebraic variety of dimension $n.$ For sufficiently general numbers $a_{ij}\in k$ the mapping

[TABLE]

is finite. $\Box$

In the sequel for a given ideal $I\subset k[x_{1},...,x_{n}]$ by $V(I)$ we mean the set of algebraic zeros of $I$, i.e., the zeroes of $I$ in $\hbox{\tenmsy K}^{n}$, where 𝕂 is an algebraic closure of $k.$ Now we can formulate our first main result:

Theorem 3**.**

Let $k$ be an infinite field and let $f_{1},\dots,f_{s}\in k[x_{1},\dots,x_{n}]$ be polynomials such $\deg f_{i}=d_{i}$ where $d_{1}\geq d_{2}...\geq d_{s}.$ Assume that $I=(f_{1},\dots,f_{s})\in k[x_{1},\dots,x_{n}]$ is a non-zero ideal, such that $V(I)$ has dimension $q\geq 0$. If we take a sufficiently general system of coordinates $(x_{1},\dots,x_{n})$, then there exist polynomials $g_{j}\in k[x_{1},\dots,x_{n}]$ and a non-zero polynomial $\phi(x)\in k[x_{1},...,x_{q+1}]$ such that

(a) $\deg g_{j}f_{j}\leq d_{s}\prod^{n-q-1}_{i=1}d_{i},$

(b) $\phi(x)=\sum^{s}_{j=1}g_{j}f_{j}.$

Proof. Let 𝕂 be the algebraic closure of $k.$ Take $F_{n-q}=f_{s}$ and $F_{i}=\sum_{j=i}^{s}\alpha_{ij}f_{j}$ for $i=1,...,n-q-1,$ where $\alpha_{ij}\in k$ are sufficiently general. Take $J=(F_{1},...,F_{n-q}).$ Then $\deg F_{n-q}=d_{s}$ and deg $F_{i}=d_{i}$ for $i=1,...,n-q-1.$ Moreover, $V(J)$ has pure dimension $q$ and $J\subset I.$ The mapping

[TABLE]

is a (non-closed) embedding outside the set $V(J)\times\mathbb{K}$. Take $\Gamma={\rm cl}(\Phi(\hbox{\tenmsy K}^{n}\times\hbox{\tenmsy K})).$ Let $\pi:\Gamma\to\hbox{\tenmsy K}^{n+1}$ be a generic projection defined over the field $k.$ Define $\Psi:=\pi\circ\Phi(x,z).$ By Lemma 2 we can assume that

[TABLE]

where $l_{1},\dots,l_{n+1}$ are generic linear form. In particular we can assume that $l_{n-q+i}$, $i=1,..,q+1$ is the variable $x_{i}$ in a new generic system of coordinates (of $\hbox{\tenmsy K}^{n}$).

Apply Theorem 1 to $\hbox{\tenmsy L}=k(z),$ and to the polynomials $\Psi_{1},\dots,\Psi_{n+1}\in\hbox{\tenmsy L}[x]$. Thus there exists a non-zero polynomial $W(T_{1},\dots,T_{n+1})\in\hbox{\tenmsy L}[T_{1},\dots,T_{n+1}]$ such that

[TABLE]

where $k=n-q.$ Since the coefficients of $W$ are in $k(z)$, there is a non-zero polynomial $\tilde{W}\in k[T_{1},\dots,T_{n+1},Y]$ such that

(a) $\tilde{W}(\Psi_{1}(x,z),\dots,\Psi_{n+1}(x,z),z)=0,$

(b) ${\deg}_{T}\tilde{W}(T_{1}^{d_{1}},T_{2}^{d_{2}},\dots,T_{k}^{d_{k}},T_{k+1},\dots,T_{n+1},Y)\leq d_{s}\prod^{n-q-1}_{j=1}d_{j},$ where ${\deg}_{T}$ denotes the degree with respect to the variables $T=(T_{1},\dots,T_{n+1}).$

Note that the mapping $\Psi=(\Psi_{1},\dots,\Psi_{n+1}):\hbox{\tenmsy K}^{n}\times\hbox{\tenmsy K}\to\hbox{\tenmsy K}^{n+1}$ is locally finite outside the set $V(J)\times\hbox{\tenmsy K}$. Consider $\hbox{\tenmsy K}^{n+1}$ as a product $\hbox{\tenmsy K}^{n-q}\times\hbox{\tenmsy K}^{q+1}$, and let us consider in this product coordinates $(y_{q+2},...,y_{n+1},y_{1},...,y_{q+1}).$ Hence $\Psi$ restricted to $V(J)\times\hbox{\tenmsy K}$ coincides with the mapping: $(x,z)\mapsto(l_{1}(x),...,l_{n-q}(x),x_{1},...,x_{q+1})$ (recall that we consider a new generic system of coordinates). Let $\phi^{\prime}=0$ describes the image of the projection

[TABLE]

Put $S=\{T\in\hbox{\tenmsy K}^{n+1}:\phi^{\prime}(T)=0\}$. Hence $V(J)\times\hbox{\tenmsy K}$ is contained in $\Psi^{-1}(S).$ Consequently the mapping $\Psi$ is proper outside the hypersurface $S$ and thus the set of non-properness of the mapping $\Psi$ is contained in the $S.$

Since the mapping $\Psi$ is finite outside $S$, for every $H\in k[x_{1},\dots,x_{n},z]$ there is a minimal polynomial $P_{H}(T,Y)\in k[T_{1},\dots,T_{n+1}][Y]$ such that $P_{H}(\Psi_{1},\dots,\Psi_{n+1},H)=\sum^{r}_{i=0}b_{i}(\Psi_{1},\dots,\Psi_{n+1})H^{r-i}=0$ and the coefficient $b_{0}$ satisfies $\{T:b_{0}(T)=0\}\subset S$. In particular $b_{0}$ depends only on variables $x_{1},...,x_{q+1}.$ Moreover, $P_{H}$ describes a hypersurface given by parametric equation $(\Psi_{1},...,\Psi_{n+1},H)$ and by Gröbner base computation we see that we can assume $P_{H}(T,Y)\in k[T_{1},\dots,T_{n+1}][Y]$. Now set $H=z.$

We have

[TABLE]

and consequently we obtain the equality $b_{0}(x_{1},...,x_{q+1})+\sum_{i=1}^{n-q}F_{i}g_{i}=0,$ where deg $F_{i}g_{i}\leq d_{s}\prod^{n-q-1}_{j=1}d_{j}$. Set $\phi=b_{0}$. By the construction the polynomial $\phi$ has zeros only on the image of the projection

[TABLE]

$\Box$

Remark 4**.**

Simple application of the Bezout theorem shows that our bound on the degree of $\phi$ is sharp.**

Corollary 5**.**

Let $k$ and $I$ and system of coordinates be as above. If $V(I)$ has pure dimension $q$ and $I$ has not embedded components, then there is a polynomial $\phi_{1}\in k[x_{1},...,x_{q+1}]$ which describes the image of the projection

[TABLE]

such that

(a) $\phi_{1}\in I$,

(b) $\deg\phi_{1}\leq d_{s}\prod^{n-q-1}_{i=1}d_{i}.$

Proof.

The set $V(J)=q$ has pure dimension $q$. Consequently $\pi(V(J))$ and $\pi(V(I))$ are hypersurfaces. Moreover, by Gröbner bases computation the set $\pi(V(I))$ is described by a polynomial $\psi$ from $k[x_{1},...,x_{q+1}].$ Let $\phi$ be a polynomial as above which vanishes exactly on $\pi(V(J)).$ Let $\phi_{1}$ be a product of all irreducible factors of $\phi$ (over the field $k$) which divides $\psi.$ Hence $\phi=\phi_{1}\phi_{2},\ \phi_{1},\phi_{2}\in k[x_{1},...,x_{q+1}]$, where $\phi_{2}$ does not vanish on any component of $V(I).$ Let $I=\bigcap^{r}I_{s}$ be a primary decomposition of $I$. Consequently $\phi_{1}\in I_{j}$ for every $s$ (by properties of primary ideals) and consequently $\phi_{1}\in I.$ But $\phi_{1}$ describes the image of the projection

[TABLE]

∎

3. A deformation argument

In this section, we improve Theorem 3 by releasing the necessity of a generic change of coordinates, so conditions (a) and (b) will be satisfied in the initial system of coordinates.

Theorem 6**.**

Let $k$ be an infinite field and let $f_{1},\dots,f_{s}\in k[x_{1},\dots,x_{n}]$ be polynomials such $\deg f_{i}=d_{i}$ where $d_{1}\geq d_{2}\geq...\geq d_{s}.$ Assume that $I=(f_{1},\dots,f_{s})\subset k[x_{1},\dots,x_{n}]$ is a non zero ideal, such that $V(I)$ has dimension $q\geq 0$. There exist polynomials $g_{j}\in k[x_{1},\dots,x_{n}]$ and a non-zero polynomial $\phi\in k[x_{1},...,x_{q+1}]$ such that

(a) $\deg g_{j}f_{j}\leq d_{s}\prod^{n-q-1}_{i=1}d_{i},$

(b) $\phi=\sum^{s}_{j=1}g_{j}f_{j}$.

Proof.

We use Theorem 3, but over the field $\hbox{\tenmsy L}=k(t)$. We consider a new generic change of coordinates using generic values $a_{i,j}$ in the infinite field $k$, together with the inverse change of coordinates

[TABLE]

where $b_{i,j}(t)\in k[t].$

As in the proof of Theorem 3, we obtain some polynomials $G_{j}\in\hbox{\tenmsy L}[X_{1},\dots,X_{n}]$ and a non-zero polynomial $b_{0}\in\hbox{\tenmsy L}[X_{n-q},...,X_{n}]$ such that, after chasing the denominators,

[TABLE]

where $b_{0}(X,t),G_{j}(X,t),\bar{F_{j}}(X,t)\in k[t][X_{1},...,X_{n}].$

We cannot just simplify this equality by $t$ and then set $t=0$, because we cannot exclude the possibility that there will be a remaining factor $t^{p}$ in the left hand side with $p$ strictly positive. To rule out this possibility, we need to perform several reduction steps. Consider the sub-module $M=\{H(x)=(H_{1}(x),\dots,H_{n-q}(x))\}$ of $k[x]^{n-q}$ formed by the relations (first syzygies) between the polynomials $F_{1}(x),\dots,F_{n-q}(x)$. To each element $H(x)$ in $M$ corresponds via the change of coordinates a relation $\bar{H}(X,t)$ between the polynomials $\bar{F}_{1}(X,t),\dots,\bar{F}_{n-q}(X,t)$, such that $\bar{H}(X,t)-{H}(X)$ is divisible by $t$. Re-writing in $(x,t)$, we obtain that

[TABLE]

We may assume that in the previous equality $b_{0}(X,t)$ has the form $b_{0}(X,t)=t^{p}(\phi(x)+t\phi_{1}(x,t))$ ; notice that the $x-$ degree of $\phi(x)$ is bounded by the $X-$degree of $b_{0}(X,t))$. Each reduction step will produce a similar equality (with the same degree in $x$ bounds) but with a strictly smaller power $p$.

Assume $p>0$ and let $t=0$, we obtain a non trivial relation $0=\sum^{n-q}_{j=1}G_{j}(x,0)F_{j}(x)$, hence $H=(G_{1}(x,0),\dots,G_{n-q}(x,0))$, a non trivial element of $M$. Notice that the $x-$ degree of $G_{j}(x,0)$ is bounded by the $X-$degree of $G_{j}(X,t)$. To which we associate its $\bar{H}$ as above with the same degree bound in $X$ (equivalently in $x$ by linearity) and notice that now $\sum^{n-q}_{j=1}(G_{j}(X,t)-\bar{H_{j}}(X,t))\bar{F_{j}}(X)$ vanishes for $t=0$, hence admits a factor $t$. We can simplify the two sides of the previous equality by $t$ and obtain

$t^{p-1}(\phi(x)+t\phi_{1}(x,t))=\sum^{n-q}_{j=1}(G_{j}(X,t)-\bar{H_{j}}(X,t))\bar{F_{j}}(X,t).$

After at most $p$ such reduction steps, we get rid of the initial factor $t^{p}$ and setting $t=0$, we obtain the announced equality with the announced bounds. ∎