A short proof for the parameter continuation theorem

TL;DR

This paper provides a concise proof of the parameter continuation theorem, fundamental in polynomial homotopy continuation, using Gr"obner bases, and introduces a method for computing discriminants.

Contribution

It offers a new, shorter proof of the parameter continuation theorem and a novel approach to computing discriminants via Gr"obner bases.

Findings

Short proof of the parameter continuation theorem

Method for computing discriminants using Gr"obner bases

Enhanced understanding of polynomial homotopy continuation

Abstract

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gr\"obner bases. Our approach gives a method for computing discriminants.

Equations

$$\mathbb{C}[\mathbf{x},\mathbf{p}]:=\mathbb{C}[x_1,\dots ,x_n,p_1,\dots ,p_k].$$

$${\mathbb{C}}^k\mapsto \mathbb{C}[\mathbf{x}{]}^{\times n},\mathbf{p}\mapsto F(\mathbf{x};\mathbf{p})=\left[\begin{array}{c}{f}_1(\mathbf{x};\mathbf{p})\\ ⋮\\ f_n(\mathbf{x};\mathbf{p})\end{array}\right]$$

$$N(\mathbf{q}):=\#\{\mathbf{x}\in {\mathbb{C}}^n\mid x\text{ is a regular zero of }F(\mathbf{x};\mathbf{q})\}.$$

$$F(\mathbf{x};\mathbf{p})=\left[\begin{array}{c}(x_1-a)\cdot (x_1-1)\cdot \gamma (\mathbf{x})\\ (x_2-3)\cdot (x_2-4{)}^2\cdot \gamma (\mathbf{x})\end{array}\right],$$

$$I:=\langle f_{1},\ldots,f_{n}\rangle\quad\text{and}\quad J:=\big{\langle}h\rangle.$$

$$\mathbf{V}(I:J^{\infty })=\overline{\mathbf{V}(I)\setminus \mathbf{V}(J)};$$

$${\varphi }_{\mathbf{q}}:\mathbb{C}[\mathbf{x},\mathbf{p}]\to \mathbb{C}[\mathbf{x}],f(\mathbf{x};\mathbf{p})\mapsto f(\mathbf{x};\mathbf{q}).$$

$$I_{\mathbf{q}}:={\varphi }_{\mathbf{q}}(I)\text{and}{J}_{\mathbf{q}}:={\varphi }_{\mathbf{q}}(J).$$

$$N(\mathbf{q})=\#\mathbf{V}(I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty }).$$

$$x_1>\cdots >x_n>p_1>\cdots >p_k.$$

$$I:J^{\infty }=(I+K)\cap \mathbb{C}[\mathbf{x},\mathbf{p}].$$

$${\varphi }_{\mathbf{q}}(G)=\{{\varphi }_{\mathbf{q}}(g_i)\mid g_i\notin \mathbb{C}[\mathbf{p}]\}$$

$$K:=⟨1-y\cdot h⟩.$$

$$(I+K)\cap \mathbb{C}[\mathbf{p}]=\{0\}.$$

$$\Delta :=\{\mathbf{q}\in {\mathbb{C}}^k\mid c_1(\mathbf{q})\cdots c_r(\mathbf{q})=0\}.$$

$${\varphi }_{\mathbf{q}}(I+K)={\varphi }_{\mathbf{q}}(I)+{\varphi }_{\mathbf{q}}(K)={\varphi }_{\mathbf{q}}(I)+(1-y\cdot {\varphi }_{\mathbf{q}}(h)).$$

$${\varphi }_{\mathbf{q}}(G)\cap \mathbb{C}[\mathbf{x}]={\varphi }_{\mathbf{q}}(\overline{G})\cap \mathbb{C}[\mathbf{x}].$$

$$\overline{G}=\{a{x}^2+bx+c,(4ac-b^2)y+2xa+b,yxb+2yc+x,2yxa+yb-1\}.$$

$$\Delta =\{\mathbf{q}=(a,b,c)\in {\mathbb{C}}^3\mid ab(4ac-b^2)=0\}.$$

$$g_3$$

$$+(-a^4-16a^2-145)x_1+a^5+16a^3+64a+81$$

$$g_4$$

$$+(-a^3-a^2-17a-17)x_1+a^4+a^3+17a^2+17a-81$$

$$\Delta =\{a\in \mathbb{C}\mid (a^2+8)(a-1)=0\}$$

Full text

University Osnabrück, Fachbereich Mathematik/Informatik Albrechtstr. 28a, 49076 Osnabrück, Germany. Both authors are supported by the Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 445466444

1. Introduction

A central task in many applications is solving a system of polynomial equations. One approach to solving such systems is polynomial homotopy continuation. To explain the basic idea we consider the polynomial ring

[TABLE]

We interpret $\mathbf{x}$ as variables and $\mathbf{p}$ as parameters.

Let $f_{1}(\mathbf{x};\mathbf{p}),\ldots,f_{n}(\mathbf{x};\mathbf{p})\in\mathbb{C}[\mathbf{x},\mathbf{p}]$. We call the image of the polynomial map

[TABLE]

a family of polynomial systems. That is, a family $\mathcal{F}=\{F(\mathbf{x};\mathbf{p})\mid\mathbf{p}\in\mathbb{C}^{k}\}$ consists of $n$ polynomials in $n$ variables $\mathbf{x}$ with $k$ parameters $\mathbf{p}$.

Let $F(\mathbf{x})$ be a system of $n$ polynomials in $n$ variables. The idea in polynomial homotopy continuation is to find a family $\mathcal{F}$ and parameters $\mathbf{q}_{1},\mathbf{q}_{2}\in\mathbb{C}^{k}$ with the following properties: $F(\mathbf{x})=F(\mathbf{x};\mathbf{q}_{1})$ and $G(\mathbf{x})=F(\mathbf{x};\mathbf{q}_{2})$ is another system whose solutions can be computed or are known. One then defines the parameter homotopy $H(\mathbf{x},t):=F(\mathbf{x};(1-t)\mathbf{q}_{1}+t\mathbf{q}_{2})$ and tracks the zeros of $H(\mathbf{x},t)$ from $t=1$ to $t=0$. This means, given $\mathbf{x}\in\mathbb{C}^{n}$ with $G(\mathbf{x})=0$, we use numerical algorithms to solve the Davidenko ODE \big{(}\tfrac{\mathrm{d}}{\mathrm{d}\mathbf{x}}H(\mathbf{x},t)\big{)}\ \dot{\mathbf{x}}+\tfrac{\mathrm{d}}{\mathrm{d}t}H(\mathbf{x},t)=0 [3, 4] for the initial value $\mathbf{x}(0)=\mathbf{x}$. In this setting, $G(\mathbf{x})$ is called the start system and $F(\mathbf{x})$ is called the target system. For more on the theory of polynomial homotopy continuation we refer to the textbook of Sommese and Wampler [10] or the overview article [1].

The Parameter Continuation Theorem by Morgan and Sommese [9] is the theoretical foundation of polynomial homotopy continuation. It implies that the initial value problem above is well posed for almost all parameters. Recall that a zero $\mathbf{x}$ of $F(\mathbf{x};\mathbf{q})$ is called regular if the Jacobian determinant \det\big{(}\tfrac{\partial f_{i}}{\partial x_{j}}\big{)}_{1\leq i,j\leq n} at $(\mathbf{x},\mathbf{q})$ does not vanish.

Theorem 1.1** (The Parameter Continuation Theorem).**

Let $\mathcal{F}$ be a family of polynomial systems that consists of systems of $n$ polynomials $F(\mathbf{x};\mathbf{p})$ in $n$ variables $\mathbf{x}$ depending on $k$ parameters $\mathbf{p}$. For $\mathbf{q}\in\mathbb{C}^{k}$ denote

[TABLE]

Let $N:=\sup_{\mathbf{q}\in\mathbb{C}^{k}}N(\mathbf{q})$. Then, $N<\infty$ and there exists a proper algebraic subvariety $\Delta\subsetneq\mathbb{C}^{k}$, called discriminant, such that $N(\mathbf{q})=N$ for all $\mathbf{q}\not\in\Delta$.

Example**.**

We consider two examples.

1. a)

The space $\mathcal{F}=\{ax^{2}+bx+c\mid a,b,c\in\mathbb{C}\}$ of univariate quadratic polynomials is a family. Here, there are 3 parameters $\mathbf{p}=(a,b,c)$ and we have $N=2$. 2. b)

Consider the family of polynomials with one parameter $\mathbf{p}=a$ defined by

[TABLE]

where $\gamma(\mathbf{x})=x_{1}^{2}+x_{2}^{2}-1$. In this case, $N=2$. Moreover, for all $\mathbf{q}\in\mathbb{C}$ the zero set of $F(\mathbf{x};\mathbf{q})$ contains a curve and at least one singular point.

A proof for the Parameter Continuation Theorem can also be found in the textbook [10]. The proofs in [9, 10] rely on the theory of holomorphic vector bundles. In this short note we give an alternative proof using Gröbner bases.

2. Gröbner bases, Saturation and Parameterized Ideals

For the proof of the Parameter Continuation Theorem we use Gröbner bases of ideals with parameters. Let $f_{1}(\mathbf{x};\mathbf{p}),\ldots,f_{n}(\mathbf{x};\mathbf{p})$ be polynomials as in (1) and denote by h(\mathbf{x};\mathbf{p}):=\det\big{(}\tfrac{\partial f_{i}}{\partial x_{j}}\big{)} the Jacobian determinant. We consider two ideals

[TABLE]

The saturation of $I$ by $J$ is the ideal $I:J^{\infty}:=\{f\in\mathbb{C}[\mathbf{x},\mathbf{p}]\mid\exists\ell>0:f\cdot h^{\ell}\in I\}.$ Saturation corresponds to removing components on the level of varieties: we have

[TABLE]

see, e.g., [2, Chapter 4 §4, Corollary 11].

Let now $\mathbf{q}\in\mathbb{C}^{k}$ be fixed and define the surjective ring homomorphism

[TABLE]

We consider the two ideals

[TABLE]

The Implicit Function Theorem implies that a regular zero of $F(\mathbf{x};\mathbf{q})$ is an isolated point in $\mathbf{V}(I_{\mathbf{q}})$. Suppose $N(\mathbf{q})=\infty$. Then, $\mathbf{V}(I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty})=\overline{\mathbf{V}(I_{\mathbf{q}})\setminus\mathbf{V}(J_{\mathbf{q}})}$ would be of positive dimension contradicting that regular zeros are isolated. Hence, $N(\mathbf{q})<\infty$. A finite union of points is Zariski closed, so $\mathbf{V}(I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty})$ is the set of regular zeros of $F(\mathbf{x};\mathbf{q})$. This implies

[TABLE]

The basic idea for our proof of the Parameter Continuation Theorem is to show that $N(\mathbf{q})$ is the degree of the projection $\mathbf{V}(I:J^{\infty})\to\mathbb{C}^{k}$, which maps $(\mathbf{x},\mathbf{q})$ to the parameter $\mathbf{q}$. We show that on the level of ideals we have $\phi_{\mathbf{q}}(I:J^{\infty})=I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty}$. One inclusion is straight-forward: suppose that $f\in I:J^{\infty}$ with $f\cdot h^{\ell}\in I$. Then, $\phi_{\mathbf{q}}(f)^{\ell}\cdot\phi_{\mathbf{q}}(h)=\phi_{\mathbf{q}}(f^{\ell}\cdot h)\in I_{\mathbf{q}}$. The other inclusion, however, does not hold in general. Our main contribution is Lemma 2.4, where we show that it holds for a general $\mathbf{q}$.

In the following, we consider the polynomial ring $\mathbb{C}[\mathbf{x},\mathbf{p}]$ with the lex order

[TABLE]

We recall two propositions related to elimination and saturation of ideals with parameters.

Proposition 2.1**.**

Let $I\subset\mathbb{C}[\mathbf{x},\mathbf{p}]$ be an ideal and $J=\langle h\rangle$ be a principal ideal. Let $y$ be an additional variable and $K:=\langle 1-y\cdot h\rangle.$ Then,

[TABLE]

Furthermore, if we augment the lex order (4) by letting $y$ be the largest variable and let $G$ be a Gröbner basis of $I+K$ relative to this order, then $G\cap\mathbb{C}[\mathbf{x},\mathbf{p}]$ is a Gröbner basis of $I:J^{\infty}$.

Proof.

See [2, Chapter 4 §4, Theorem 14]. ∎

Proposition 2.2**.**

Consider an ideal $L\subset\mathbb{C}[\mathbf{x},\mathbf{p}]$ and let $G=\{g_{1},\ldots,g_{s}\}$ be a Gröbner basis for $L$ relative to lex order in (4). For $1\leq i\leq s$ with $g_{i}\not\in\mathbb{C}[\mathbf{p}]$, write $g_{i}$ in the form $g_{i}=c_{i}(\mathbf{p})\mathbf{x}^{\alpha_{i}}+h_{i}$, where all terms of $h_{i}$ are strictly smaller than $\mathbf{x}^{\alpha_{i}}$. Let $\mathbf{q}\in\mathbf{V}(L\cap\mathbb{C}[\mathbf{p}])\subseteq\mathbb{C}^{k}$, such that $c_{i}(\mathbf{q})\neq 0$ for all $g_{i}\not\in\mathbb{C}[\mathbf{p}]$. Then,

[TABLE]

is a Gröbner basis for the ideal $\phi_{\mathbf{q}}(L)\subset\mathbb{C}[\mathbf{x}]$.

Proof.

See [2, Chapter 4 §7, Theorem 2]. ∎

Remark 2.3**.**

Weispfenning [12] proved the existence of a Comprehensive Gröbner Basis (CGB). In the notation of Proposition 2.2 this is a Gröbner basis $G$ of $L$, such that $\phi_{\mathbf{q}}(G)$ is a Gröbner basis for $\phi_{\mathbf{q}}(L)$ for all $\mathbf{q}\in\mathbb{C}^{k}$. We also refer the reader to the following works of Weispfenning, Montes, Kapur and others [13, 7, 8, 6], which considerably improved the construction and optimized the algorithm for computing a CGB. Nevertheless, for our purposes, Proposition 2.2 is enough. In addition, the condition of non-vanishing leading coefficients is essential for us.

The next lemma is our main contribution.

Lemma 2.4**.**

Let $I\subset\mathbb{C}[\mathbf{x},\mathbf{p}]$ be an ideal and $J=\langle h\rangle$ be a principal ideal, such that $(I:J^{\infty})\cap\mathbb{C}[\mathbf{p}]=\{0\}$. Let $G=\{g_{1},\ldots,g_{s}\}$ be a Gröbner basis of $I:J^{\infty}$ relative to the lex order (4). There is a proper subvariety $\Delta\subsetneq\mathbb{C}^{k}$ such that for all $\mathbf{q}\not\in\Delta$ the set $\{\phi_{\mathbf{q}}(g_{1}),\ldots,\phi_{\mathbf{q}}(g_{s})\}$ is a Gröbner basis for $\phi_{\mathbf{q}}(I):\phi_{\mathbf{q}}(J)^{\infty}$ and none of the leading terms of $g_{1},\ldots,g_{s}$ vanish when evaluated at $\mathbf{q}$.

In particular, $\phi_{\mathbf{q}}(I:J^{\infty})=\phi_{\mathbf{q}}(I):\phi_{\mathbf{q}}(J)^{\infty}$ for all $\mathbf{q}\not\in\Delta$.

Proof.

Let $y$ be an additional variable and, as in Proposition 2.1, denote

[TABLE]

By Proposition 2.1, we have $I:J^{\infty}=(I+K)\cap\mathbb{C}[\mathbf{x},\mathbf{p}]$. Since $(I:J^{\infty})\cap\mathbb{C}[\mathbf{p}]=\{0\}$, we therefore have

[TABLE]

In particular, $\mathbf{V}((I+K)\cap\mathbb{C}[\mathbf{p}])=\mathbb{C}^{k}$ and we may therefore apply Proposition 2.2 to $I+K$ without putting any restrictions on $\mathbf{q}$.

As in Proposition 2.2, we augment the lex order (4) by letting $y$ be the largest variable. Let $\overline{G}:=\{g_{1},\ldots,g_{r}\}$ be a Gröbner basis of $I+K$ relative to this order. It follows from (5) that we have $g_{1},\ldots,g_{r}\notin\mathbb{C}[\mathbf{p}]$. We write each $g_{i}$ in the form $g_{i}=c_{i}(\mathbf{p})y^{\beta}\mathbf{x}^{\alpha_{i}}+h_{i}$, where all terms of $h_{i}$ are strictly smaller than $y^{\beta}\mathbf{x}^{\alpha_{i}}$, and define the hypersurface

[TABLE]

In the following, let $\mathbf{q}\in\mathbb{C}^{k}\setminus\Delta$. By Proposition 2.2, $\phi_{\mathbf{q}}(\overline{G})=\{\phi_{\mathbf{q}}(g_{1}),\ldots,\phi_{\mathbf{q}}(g_{r})\}$ is a Gröbner basis for

[TABLE]

Without restriction, the first $s\leq r$ elements in $\overline{G}$ are those that do not depend on $y$. We denote $G:=\{g_{1},\ldots,g_{s}\}=\overline{G}\cap\mathbb{C}[\mathbf{x},\mathbf{p}]$. It follows from Proposition 2.1 that $G$ is a Gröbner basis of $I:J^{\infty}$. Because $\mathbf{q}\not\in\Delta$, none of the leading terms in $\overline{G}$ when evaluated at $\mathbf{q}$ vanish. Consequently,

[TABLE]

Therefore, $\phi_{\mathbf{q}}(G)=\{\phi_{\mathbf{q}}(g_{1}),\ldots,\phi_{\mathbf{q}}(g_{s})\}$ is a Gröbner basis of $\phi_{\mathbf{q}}(I):\phi_{\mathbf{q}}(J)^{\infty}$ by Proposition 2.1. ∎

Example**.**

We illustrate Lemma 2.4 using the two examples from the introduction.

1. a)

For $\mathcal{F}=\{ax^{2}+bx+c\mid a,b,c\in\mathbb{C}\}$ we have $I=\langle ax^{2}+bx+c\rangle$ and $J=\langle 2ax+b\rangle$. We first compute a Gröbner basis for $I:J^{\infty}$ using Macaulay2 [5]:

⬇

R = QQ[x, a, b, c, MonomialOrder => Lex];

f = a * x^2 + b * x + c; h = 2 * a * x + b;

I = ideal {f};

J = ideal {h};

S = saturate(I, J);

G = gens gb S

This yields $G=\{ax^{2}+bx+c\}$. Now we consider two sets of parameters. First, $\mathbf{q}_{1}=(1,3,2)$ and then $\mathbf{q}_{2}=(1,-2,1)$.

⬇

Iq = sub(I, {a=>1, b=>3, c=>2});

Jq = sub(J, {a=>1, b=>3, c=>2});

Sq = saturate(Iq, Jq); Gq = gens gb Sq

This gives us the Gröbner basis $\{\phi_{\mathbf{q}_{1}}(ax^{2}+bx+c)\}=\{x^{2}+3x+2\}$ for $\phi_{\mathbf{q}_{1}}(I)\colon\phi_{\mathbf{q}_{1}}(J)^{\infty}$. On the other hand, $\phi_{\mathbf{q}_{2}}(ax^{2}+bx+c)=x^{2}-2x+1=(x-1)^{2}$, so that in this case, $\phi_{\mathbf{q}_{2}}(I)\colon\phi_{\mathbf{q}_{2}}(J)^{\infty}=\langle 1\rangle$. 2. b)

For the second example we also compute Gröbner basis for $I:J^{\infty}$ using Macaulay2 [5]:

⬇

R = QQ[x1, x2, a, MonomialOrder => Lex];

gamma = x1^2 + x2^2 - 1;

f1 = (x1 - a) * (x1 - 1) * gamma;

f2 = (x2 - 3) * (x2 - 4)^2 * gamma;

h = diff(x1, f1) * diff(x2, f2) - diff(x1, f2) * diff(x2, f1);

I = ideal {f1, f2};

J = ideal {h};

S = saturate(I, J);

G = gens gb S

This yields the Gröbner basis $G=\{x_{2}-3,\ x_{1}^{2}-ax_{1}-x_{1}+a\}$. We compute the saturation for $\mathbf{q}_{1}=1$ and $\mathbf{q}_{2}=2$. In the first case:

⬇

Iq = sub(I, {a=>1});

Jq = sub(J, {a=>1});

Sq = saturate(Iq, Jq)

gives the ideal $\phi_{\mathbf{q}_{1}}(I)\colon\phi_{\mathbf{q}_{1}}(J)^{\infty}=\langle 1\rangle$. This is because for $a=1$ the two regular zeros $(a,3)$ and $(1,3)$ come together to form a singular zero. On the other hand, $\phi_{\mathbf{q}_{2}}(I)\colon\phi_{\mathbf{q}_{2}}(J)^{\infty}$ has Gröbner basis $\phi_{\mathbf{q}_{2}}(G)=\{x_{2}-3,\ x_{1}^{2}-3x_{1}+2\}$.

We now prove the Parameter Continuation Theorem.

Proof of Theorem 1.1.

If $N=0$, then no system in $\mathcal{F}$ has regular zeros. In this case, the statement is true.

We now assume $N>0$. By (2), $\mathbf{V}(I:J^{\infty})$ contains all $(\mathbf{x},\mathbf{q})\in\mathbb{C}^{n}\times\mathbb{C}^{k}$ such that $\mathbf{x}$ is a regular zero of $F(\mathbf{x};\mathbf{q})$. Since $N>0$, we therefore have $\mathbf{V}(I:J^{\infty})\neq\emptyset$. Let $(\mathbf{x},\mathbf{q})\in\mathbf{V}(I:J^{\infty})$. The Implicit Function Theorem implies that there is a Euclidean open neighbourhood $U$ of $\mathbf{q}$ such that $F(\mathbf{x};\mathbf{q})$ has regular zeros for all $\mathbf{q}\in U$. Consequently, $(I:J^{\infty})\cap\mathbb{C}[\mathbf{p}]=\{0\}$, so we can apply Lemma 2.4.

As before, we denote $I_{\mathbf{q}}=\phi_{\mathbf{q}}(I)$ and $J_{\mathbf{q}}=\phi_{\mathbf{q}}(J)$. By Lemma 2.4, we have $\phi_{\mathbf{q}}(I:J^{\infty})=I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty}$ for general $\mathbf{q}$. This implies that $\mathbf{V}(I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty})$ is the fiber of the projection $\mathbf{V}(I:J^{\infty})\to\mathbb{C}^{k},\,(\mathbf{x},\mathbf{q})\mapsto\mathbf{q}$ for general $\mathbf{q}$. Therefore, $\#\mathbf{V}(I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty})$ is the degree of this projection, hence constant and maximal for general $\mathbf{q}$. The statement now follows from $N(\mathbf{q})=\#\mathbf{V}(I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty})$, which we showed in (3). ∎

Remark 2.5**.**

The following yields a more explicit proof for Theorem 1.1, not working with degrees of projections.

The idea is to show that for general $\mathbf{q}$ the Gröbner bases of $I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty}$ all have the same number of standard monomials. Recall that one calls a monomial $\mathbf{x}^{\alpha}$ a standard monomial of an ideal $L$ (relative to a monomial order), if it is not in $\mathrm{LT}(L)$, the ideal of leading terms in $L$. Let $\mathcal{B}$ be the set of standard monomials of $L$ Then, $\mathcal{B}$ is finite if and only if $\mathbf{V}(L)$ is finite, and $\#\mathcal{B}$ equals the number of points in $\mathbf{V}(L)$ counting multiplicities (see [11, Proposition 2.1]).

Now, let $G=\{g_{1},\ldots,g_{s}\}$ be a Gröbner basis of $I:J^{\infty}$ relative to the lex order from (4). By Lemma 2.4, there is a proper algebraic subvariety $\Delta\subsetneq\mathbb{C}^{k}$ such that $\phi_{\mathbf{q}}(G)=\{\phi_{\mathbf{q}}(g_{1}),\ldots,\phi_{\mathbf{q}}(g_{s})\}$ is a Gröbner basis for $I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty}$ for all $\mathbf{q}\not\in\Delta$. Moreover, none of the leading terms of $g_{1},\ldots,g_{s}$ vanish when evaluated at $\mathbf{q}\not\in\Delta$. This implies that the leading monomials of $I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty}$ are constant on $\mathbb{C}^{k}\setminus\Delta$. Thus, if $\mathcal{B}_{\mathbf{q}}$ denotes the set of standard monomials of $I_{\mathbf{q}}:J_{\mathbf{q}}^{\infty}$, also $\mathcal{B}_{\mathbf{q}}$ is constant on $\mathbb{C}^{k}\setminus\Delta$. On the other hand, $N(\mathbf{q})=\#\mathcal{B}_{\mathbf{q}}$ by (3) and the fact that regular zeros have multiplicity one. This shows that $N(\mathbf{q})$ is constant on $\mathbb{C}^{k}\setminus\Delta$. The Implicit Function Theorem implies that for all $\mathbf{q}\in\mathbb{C}^{k}$ there exists a Euclidean neighbourhood $U$ of $\mathbf{q}$ such that $N(\mathbf{q})\leq N(\mathbf{q}^{\prime})$ for all $\mathbf{q}^{\prime}\in U$. Since $\Delta$ is a proper subvariety of $\mathbb{C}^{k}$ and thus lower-dimensional, we have $N(\mathbf{q})=N<\infty$ for $\mathbf{q}\in\mathbb{C}^{k}\setminus\Delta$.

The description of the discriminant $\Delta$ in (6) leads to an algorithm for computing it: given $I=\langle f_{1},\ldots,f_{n}\rangle$, we first compute the Jacobian determinant h=\det\big{(}\tfrac{\partial f_{i}}{\partial x_{j}}\big{)}. Then, we compute a lex Gröbner basis for $I+\langle 1-y\cdot h\rangle$. The product of the leading coefficients $c_{i}(\mathbf{p})$ of this Gröbner basis gives us an equation for the discriminant. It is important to emphasize that this algorithm does not yield the smallest hypersurface $\Delta$ with the propertie in Theorem 1.1. We will see this in the next example.

Example**.**

We consider again the two examples from the introduction.

1. a)

We compute the discriminant for $\mathcal{F}=\{ax^{2}+bx+c\mid a,b,c\in\mathbb{C}\}$:

⬇

R = QQ[y, x, a, b, c, MonomialOrder => Lex];

f = a * x^2 + b * x + c; h = 2 * a * x + b;

I = ideal {f};

K = ideal {1 - y * h};

G = gens gb (I+K)

This gives us the Gröbner basis

[TABLE]

The leading terms are $c_{1}(\mathbf{p})=a$, $c_{2}(\mathbf{p})=4ac-b^{2}$, $c_{3}(\mathbf{p})=b$ and $c_{4}(\mathbf{p})=2a$. We get the discriminant

[TABLE]

Indeed, $ax^{2}+bx+c$ has less than two regular zeros if and only if $a=0$ or $4ac-b^{2}=0$. The additional factor $b$ is no contraction: we show that if $\mathbf{q}\not\in\Delta$ then $N(\mathbf{q})=N$ is maximal, but we do not show that $N(\mathbf{q})=N$ implies $\mathbf{q}\not\in\Delta$. 2. b)

In the second example, we compute the discriminant analogously:

⬇

R = QQ[y, x1, x2, a, MonomialOrder => Lex];

gamma = x1^2 + x2^2 - 1;

f1 = (x1 - a) * (x1 - 1) * gamma;

f2 = (x2 - 3) * (x2 - 4)^2 * gamma;

h = diff(x1, f1) * diff(x2, f2) - diff(x1, f2) * diff(x2, f1);

I = ideal {f1, f2};

K = ideal {1 - y * h};

G = gens gb (I+K)

The Gröbner basis we get is $\overline{G}=\{g_{1},g_{2},g_{3},g_{4}\}$ with polynomials $g_{1}=x_{2}-3$, $g_{2}=x_{1}^{2}-ax_{1}-x_{1}+a$ and

[TABLE]

The following code then finds the discriminant:

⬇

E = (entries(G))#0

P = QQ[a][y, x1, x2, MonomialOrder => Lex]

result = apply(E, t -> leadCoefficient(sub(t, P)))

factor(product result)

The result is

[TABLE]

as the leading term of $g_{3}$ is $81(a^{2}+8)^{2}(a-1)^{2}$. Indeed, the parameters for which we obtain less regular zeros than 2 are $a=1$ (in this case, $(a,3)$ is a double root) and $a=\pm\sqrt{-8}$ (in this case, $(a,3)$ lies on the circle $\gamma(\mathbf{x})=0$).