On Computing the Elimination Ideal Using Resultants with Applications to Gröbner Bases

The aim of the work presented in this paper is to study elementary relations between resultants and elimination ideals. Given an ideal $I$ in a polynomial ring with indeterminates $x_{1},\dotsc,x_{n}$, we call first elimination ideal of $I$ the intersection $I\cap\mathbb{K}[x_{2},\dotsc,x_{n}]$. Understanding such ideals is part of the so-called elimination problem, an old and central topic in polynomial algebra.

Historically the motivation for investigating such a problem comes from the polynomial systems solving and the desire to reduce a system in $n$ variables to another one involving less variables. In this context, many different tools appeared, such as resultants and Gröbner bases.

The problem of defining and investigating the notion of resultant has been considered, among others, by Sylvester, Bezout, Dixon, Macaulay and van der Waerden (see [2]). Gröbner focused on elimination ideals in [4]. A modern view of the theory of resultants was given by Gelfand, Kapranov and Zelevinski in [5]. A survey paper by Emiris and Mourrain [3] discusses determinantal representations of resultants and related computational questions.

In Section 2, we focus on the case of ideals $I$ generated by two polynomials. In this setting, it is natural to consider the resultant of the two polynomials with respect to one of the variables. We recall some well-known results in elimination theory, and provide an affine version of the result linking the variety of the resultant and the projection of the variety of the ideal $I$. The main result, Corollary 2.7, shows that, if the resultant is not identically zero, the variety of the elimination ideal and the projection of the variety of the ideal coincide.

In Section 3, we examine the relation between the multiplicity of each factor of the resultant of two polynomials, and the multiplicity of the corresponding factor in the generator of the first elimination ideal. In [6] Lazard gave a structure theorem for the minimal lexicographic Gröbner basis of a bivariate ideal generated by any number of polynomials, which reveals some of the factors of the generator of the elimination ideal. We provide examples exhibiting possible behaviour of these two multiplicities.

In this section we are going to investigate some relations between the zero set of the resultant of two polynomials, the zero set of ideal they generate, and the zero set of the first elimination ideal of the latter. The main result is Corollary 2.7.

For an ideal $I$ in $\mathbb{K}[x_{1},x_{2},\ldots,x_{n}]$ — where $\mathbb{K}$ is an algebraically closed field — we denote by $\mathcal{V}\left(I\right)$ its associated variety and by $I_{1}$ the first elimination ideal of $I$, i.e., $I_{1}=I\cap\mathbb{K}[x_{2},x_{3},\dotsc,x_{n}]$. We recall two main results on the connection between $I_{1}$ and $\mathcal{V}\left(I\right)$.

For $f_{1},\ldots,f_{m}\in\mathbb{K}[x_{1},x_{2},\ldots,x_{n}]$, we write $f_{i}$ in the form

for each $1\leq i\leq m$. Consider the projection $\pi\colon\mathbb{K}^{n}\rightarrow\mathbb{K}^{n-1}$:

Although the projection of the variety of an ideal and the variety of the elimination ideal are in general not the same, the latter is the Zariski closure of the former.

Also in the following example we see that we can have $\mathcal{V}\left(I_{1}\right)=\pi\bigl{(}\mathcal{V}\left(I\right)\bigr{)}$, independently of the dimension of $\mathcal{V}\left(h_{1},\ldots,h_{m}\right)$.

Also, not necessarily $\mathcal{V}\left(h_{1},\ldots,h_{m}\right)\subseteq\mathcal{V}\left(I_{1}\right)$ is true. Note that $\mathcal{V}\left(h_{1},\ldots,h_{m}\right)$ is not the complement of $\pi\bigl{(}\mathcal{V}\left(I\right)\bigr{)}$, but contains the complement. Moreover, the dimensions of $\mathcal{V}\left(h_{1},\ldots,h_{m}\right)$ and the complement are independent of each other.

As mentioned above, we will investigate the relation between the first elimination ideal and the resultant. We first introduce some notation concerning resultants. Let $f_{1},f_{2}\in\mathbb{K}[x_{1},x_{2},\ldots,x_{n}]$ be polynomials of degree $d_{1}$ and $d_{2}$ respectively. Think of them as elements of $\mathbb{K}[x_{2},\dotsc,x_{n}][x_{1}]$ and denote by $f_{i,j}$ the coefficient of $x_{1}^{j}$ in $f_{i}$. Recall that the resultant of $f_{1}$ and $f_{2}$ with respect to $x_{1}$ is defined as

where $\operatorname{Syl}(f_{1},f_{2})$ is the Sylvester matrix, namely

In the following, we consider the connection between the zero set of the resultant of two polynomials $f_{1}$ and $f_{2}$, and the projection of the variety of the ideal $\left\langle f_{1},f_{2}\right\rangle$. In this sense, the situation is similar to the one of the Elimination Theorem. The homogeneous version of this result is an easy consequence of the basic properties of the resultant. Hereafter we propose an affine version of it, based upon the ideas of [1, Chapter 3.6, Proposition 3].

In Section 2 we focused on the relations between the varieties associated to an ideal and its elimination ideal. Here we want to deal with the algebraic side of the question; in particular, we are interested in the relations between the multiplicities of the factors of the resultant of two polynomials and the multiplicities of the factors of the generator of the elimination ideal.

We fix an elimination order on the polynomial ring $\mathbb{K}[x_{1},x_{2},\ldots,x_{n}]$ such that $x_{i}\prec x_{1}$ for all $i\in\{2,\dotsc,n\}$. The celebrated Elimination Property of Gröbner bases asserts that if $G$ is a Gröbner basis for an ideal $I$ with respect to the fixed elimination order, then $G\cap\mathbb{K}[x_{2},x_{3},\ldots,x_{n}]$ is a Gröbner basis for the elimination ideal $I_{1}$ with respect to the same order.

Given two polynomials $f_{1}$ and $f_{2}$, we denote by $S_{12}$ their S-polynomial, which is defined as follows:

where $\operatorname{lm}$ denotes the leading monomial.

Lemma 3.1 also helps us proving the next proposition.

From now on we consider two polynomials $f_{1},f_{2}\in\mathbb{K}[x,y]$. As already mentioned, if $I=\left\langle f_{1},f_{2}\right\rangle$ and $g$ denotes the generator of the elimination ideal $I_{1}=I\cap\mathbb{K}[y]$, then the resultant $\mathcal{R}=\operatorname{res}_{x}\left(f_{1},f_{2}\right)$ is a multiple of $g$. In particular, the factors of $g$ are factors of $\mathcal{R}$. If we are given the resultant $\mathcal{R}$ and we want to recover $g$, we just need to understand what are the correct multiplicities for the factors of $g$.

Let $c\in\mathbb{K}$ be a root of $g$, and let $\mu$ and $\nu$ be the multiplicities of the factor corresponding to $c$ in $g$ and $\mathcal{R}$ respectively. Clearly $\mu\leq\nu$. We exhibit some examples of situations that can arise.

Case $\nu=1$:

Here, either the linear factor vanishing on $c$ appears in $g$ with multiplicity $1$, or it does not appear at all. Using the notation of Equation (1), such a factor appears either in $g$ or in $\operatorname{gcd}\left(h_{1},h_{2}\right)$. Hence, if $\mathcal{R}$ is squarefree, then $g=\frac{\mathcal{R}}{\operatorname{gcd}\left(h_{1},h_{2}\right)}$.

The following example shows such a situation. Let $f_{1}=xy-1$, $f_{2}=x^{2}y+y^{2}-4$. Then $\mathcal{R}=y(y^{3}-4y+1)$ and $I_{1}=\left\langle y^{3}-4y+1\right\rangle$. The value $0$ is a root of $\mathcal{R}$ of multiplicity $1$, but it is not a root of $g$. The $\gcd$ of $h_{1}$ and $h_{2}$ is $y$, and $g=\frac{\mathcal{R}}{\operatorname{gcd}\left(h_{1},h_{2}\right)}$.

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle xy-1$ $\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle x^{2}y+y^{2}-4$ $\displaystyle h_{1}$ $\displaystyle=$ $\displaystyle y$ $\displaystyle h_{2}$ $\displaystyle=$ $\displaystyle y$ $\displaystyle g$ $\displaystyle=$ $\displaystyle y^{3}-4y+1$ $\displaystyle\mathcal{R}$ $\displaystyle=$ $\displaystyle y(y^{3}-4y+1)$

Case $\nu>1$:

Here a root of $\mathcal{R}$ appears with multiplicity greater than $1$. $\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle-(y+1)(x-y-1)$ $\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle x^{2}+y^{2}-1$ $\displaystyle h_{1}$ $\displaystyle=$ $\displaystyle-(y+1)$ $\displaystyle h_{2}$ $\displaystyle=$ $\displaystyle 1$ $\displaystyle g$ $\displaystyle=$ $\displaystyle y(y+1)^{2}$ $\displaystyle\mathcal{R}$ $\displaystyle=$ $\displaystyle 2y(y+1)^{3}$

The factor $y$ in $g$ is preserved with the same multiplicity as in $\mathcal{R}$, but the multiplicity of the factor $(y+1)$ drops by $1$.

One can notice that in the case $\nu>1$ above we are in the situation covered by Remark 3.5, since $(y+1)$ and the circle have a common tangent parallel to the $x$-axis at their intersection.

The multiplicity structure of isolated points can be studied by means of the dual space of the vanishing ideal of those points. In [7], the problem of understanding the differences between the multiplicities of the factors of the resultant $\mathcal{R}$ and the generator $g$ of the elimination ideal has been addressed via dual spaces; there, the concept of directional multiplicity has been introduced to explain the exponents of the factors in ${g}$.