# Toric Sylvester forms and applications in elimination theory

Laurent Busé  Université Côte d’Azur, Inria, France  Carles Checa  National
Kapodistrian University of Athens, Athena RC, Greece

###### Abstract

In this paper, we investigate the structure of the saturation of ideals
generated by square systems of sparse homogeneous polynomials over a toric
variety $X$ with respect to the irrelevant ideal of $X$. As our main results,
we establish a duality property and make it explicit by introducing toric
Sylvester forms, under a certain positivity assumption on $X$. In particular,
we prove that toric Sylvester forms yield bases of some graded components of
$I^{\text{sat}}/I$, where $I$ denotes an ideal generated by $n+1$ generic
forms, $n$ is the dimension of $X$ and $I^{\text{sat}}$ the saturation of $I$
with respect to the irrelevant ideal of the Cox ring of $X$. Then, to
illustrate the relevance of toric Sylvester forms we provide three
consequences in elimination theory: (1) we introduce a new family of
elimination matrices that can be used to solve sparse polynomial systems by
means of linear algebra methods, including overdetermined polynomial systems;
(2) by incorporating toric Sylvester forms to the classical Koszul complex
associated to a polynomial system, we obtain new expressions of the sparse
resultant as a determinant of a complex; (3) we prove a new formula for
computing toric residues of the product of two forms.

##  1 Introduction

The elimination of variables from a system of homogeneous polynomials is
deeply connected to the saturation of ideals with respect to a certain
geometrically irrelevant ideal. Thus, the search and study of universal
generators of the saturation of an ideal generated by generic homogeneous
polynomials is an important topic in elimination theory. Actually, in the
classical literature of the previous century such universal generators were
called inertia forms by Hurwitz, Mertens, Van der Waerden and many others,
including Zariski; see the references in [Jou91, Jou97] and [Zar37]. As
examples, Jacobian determinants and resultants associated to a square
polynomial system are important inertia forms.

To be more specific, consider the ideal $I=(F_{0},\dots,F_{n})$ where $F_{i}$
is the generic homogeneous polynomial of degree $d_{i}$ in the (canonically)
graded polynomial ring $C=A[x_{0},\dots,x_{n}]$, with $A$ standing for the
universal ring of coefficients of the $F_{i}$’s. The saturation of the ideal
$I$ with respect to the ideal $\mathfrak{m}=(x_{0},\ldots,x_{n})$, which we
denote by $I^{\operatorname{sat}}=I:\mathfrak{m}^{\infty}$, is the ideal of
inertia forms. In this context, the ideal $\mathfrak{m}$ is the
(geometrically) irrelevant ideal of the projective space of dimension $n$
which is associated to $C$. The elements in $I$ being trivially inertia forms,
$I^{\operatorname{sat}}/I$ is the natural quotient to study. It turns out that
the Jacobian determinant of the $F_{i}$’s is a generator of the graded
component of $I^{\operatorname{sat}}/I$ in degre
$\delta:=d_{0}+\dots+d_{n}-(n+1)$ and their resultant is a generator of
$I^{\operatorname{sat}}/I$ in degree 0. In order to unravel the structure of
$I^{\operatorname{sat}}/I$ in degrees smaller than $\delta$, the formalism of
Sylvester forms has been introduced and studied by Jouanolou in [Jou97]. His
ideas were based on the fact that for each
$\mu=(\mu_{0},\dots,\mu_{n})\in\mathbb{N}^{n+1}$ such that
$|\mu|:=\sum_{i}\mu_{i}<\min_{i}d_{i}$, it is possible to find a decomposition
of each polynomial $F_{i}$ of the form

$F_{i}=\sum_{j}x_{j}^{\mu_{j}+1}F_{i,j},$ (1.1)

and hence to consider the determinant $\det(F_{i,j})_{0\leq i,j\leq n}$. This
latter is called a Sylvester form of the $F_{i}$’s and denoted by
$\operatorname{Sylv}_{\mu}$. Independently of the decomposition, the class of
$\operatorname{Sylv}_{\mu}$, which is denoted by $\operatorname{sylv}_{\mu}$,
gives a nonzero element in $(I^{\operatorname{sat}}/I)_{\delta-|\mu|}$ and
additionally, $(I^{\operatorname{sat}}/I)_{\delta-|\mu|}$ is a free $A$-module
which can be generated by Sylvester forms of degree $\delta-|\mu|$. This
result is a consequence of a duality property between Sylvester forms and
monomials; namely, for all $\nu<\min_{i}d_{i}$ we have an isomorphism of
$A$-modules

$(I^{\operatorname{sat}}/I)_{\delta-\nu}\simeq\operatorname{Hom}_{A}(C_{\nu},A).$

It is made explicit by the equalities

$x^{\mu^{\prime}}\operatorname{sylv}_{\mu}=\begin{cases}\operatorname{sylv}_{0}&\mu=\mu^{\prime}\\\
0&\mu\neq\mu^{\prime}\end{cases}$

where $\operatorname{sylv}_{0}$ is a generator of
$(I^{\operatorname{sat}}/I)_{\delta}$, which is equal, up to a nonzero
multiplicative constant, to the class of the Jacobian determinant of the
$F_{i}$’s; see [Jou97, §3.10].

The definition and main properties of Sylvester forms have been recently
extended to the case of $n+1$ generic multi-homogeneous polynomials, i.e. of
polynomials defining hypersurfaces over a product of projective spaces of
total dimension $n$; see [BCN22]. In this paper, we develop the theory of
Sylvester forms in the general setting of homogeneous polynomials in the
coordinate ring of a projective toric variety $X_{\Sigma}$. In addition, to
illustrate the importance of these forms in elimination theory, we also
provide applications to the construction of elimination matrices for
overdetermined polynomial systems, and to the computation of toric resultants
and toric residues. As far as we know, these applications provide new results
in the context of multi-homogeneous polynomial systems as well.

Let $k$ be an algebraically closed field and $X_{\Sigma}$ be a $n$-dimensional
projective toric variety over $k$ given by a complete fan $\Sigma$ in a
lattice $N$. Let $R$ be the homogeneous coordinate ring of $X_{\Sigma}$ over
$k$, also known as the Cox ring of $X_{\Sigma}$, which is graded using the
combinatorics of $\Sigma$; see Section 2 or [Cox95] for more details. Choosing
a $n$-dimensional cone $\sigma\in\Sigma(n)$, the variables of $R$ can be
written as $x_{1},\ldots,x_{n}$, which are associated to $\sigma$, and
$z_{1},\ldots,z_{r}$ for the remaining ones. With these notations, a
homogeneous polynomial in $X_{\Sigma}$ of degree
$\alpha\in\operatorname{Cl}(X_{\Sigma})$, the class group of $X_{\Sigma}$, is
an element in the graded component $R_{\alpha}$ of $R$ in degree $\alpha$; it
is a $k$-linear combination of monomials $x^{\mu}:=x_{1}^{\mu_{1}}\cdots
x_{n}^{\mu_{n}}z_{1}^{\mu_{n+1}}\cdots z_{r}^{\mu_{n+r}}$ of degree $\alpha$.
Now, the generic homogeneous polynomial of degree $\alpha$ is the polynomial
of the form $\sum_{x^{\mu}\in R_{\alpha}}c_{i,\mu}x^{\mu}$ where the
coefficients $c_{i,\mu}$ are seen as variables. Therefore, being given $n+1$
degrees $\alpha_{0},\ldots,\alpha_{n}$, the associated generic homogeneous
polynomial system over $X_{\Sigma}$ corresponds to the $n+1$ homogeneous
polynomials

$F_{i}=\sum_{x^{\mu}\in R_{\alpha_{i}}}c_{i,\mu}x^{\mu}\in
C=A\otimes_{k}R=A[x_{1},\dots,x_{n},z_{1},\dots,z_{r}],i=0,\ldots,n,$ (1.2)

where $A$ is the universal ring of coefficients over $k$, i.e.
$A=k[c_{i,\mu}\,:\,x^{\mu}\in R_{\alpha_{i}},i=0,\dots,n].$ We define the
ideals $I=(F_{0},\ldots,F_{n})$ and
$\mathfrak{b}=(\tilde{x}^{\sigma}\,:\,\tilde{x}^{\sigma}=\prod_{\rho\notin\sigma}x_{\rho},\,\sigma\in\Sigma(n))$,
where the latter is the irrelevant ideal of $X_{\Sigma}$. The saturation of
$I$ is thus the ideal of $C$ defined as
$I^{\operatorname{sat}}=(I:\mathfrak{b}^{\infty})$.

As the first main result of this paper, we prove the following duality
property, which is a generalization of [BCN22, Theorem A] to the case of a
projective toric variety; see Theorem 3.1. We set
$\delta=\alpha_{0}+\dots+\alpha_{n}-K_{X}\in\operatorname{Cl}(X_{\Sigma})$,
where $K_{X}$ is the anticanonical class of $X_{\Sigma}$.

######  Theorem.

Let $X_{\Sigma}$ be a projective toric variety and let
$\nu\in\operatorname{Cl}(X_{\Sigma})$. There exists a region
$\Gamma\subset\operatorname{Cl}(X_{\Sigma})$ such that if
$\delta-\nu\notin\Gamma$ then

$(I^{\operatorname{sat}}/I)_{\delta-\nu}\simeq\operatorname{Hom}_{A}((C/I)_{\nu},A).$

In cases where $(C/I)_{\nu}=C_{\nu}$, the above duality implies that
$(I^{\operatorname{sat}}/I)_{\delta-\nu}$ is a free $A$-module (see Corollary
3.1) and a natural question is to find explicit bases. This is what we provide
as the second main contribution of this paper, by introducing toric Sylvester
forms. We first prove that for suitable $\nu$ and $x^{\mu}\in R_{\nu}$, each
generic homogeneous polynomial $F_{i}$ can be decomposed into $n+1$ generic
homogeneous polynomials $(F_{ij}^{\mu})_{0\leq j\leq n}$, similarly to (1.1).
The existence of such decompositions requires a certain property on
$X_{\Sigma}$ that we introduce and call positivity property; see Theorem 2.1.
Then, from these decompositions we define toric Sylvester forms as the
determinants $\operatorname{Sylv}_{\mu}:=\det(F^{\mu}_{ij})\in
I^{\operatorname{sat}}_{\delta-\nu}$ and show that their classes in
$(I^{\operatorname{sat}}/I)_{\delta-\nu}$, denoted by
$\operatorname{sylv}_{\mu}$, are independent of the choice of decompositions.
Finally, we get the following explicit duality property; see Theorem 4.1.

######  Theorem.

Let $X_{\Sigma}$ be a projective toric variety. Then, under suitable
conditions on $\nu\in\operatorname{Cl}(X_{\Sigma})$, for any pair
$x^{\mu},x^{\mu^{\prime}}\in R_{\nu}$, we have

$x^{\mu}\operatorname{sylv}_{\mu^{\prime}}=\begin{cases}\operatorname{sylv}_{0}&\mu=\mu^{\prime}\\\
0&\text{otherwise}\end{cases}$

where $\operatorname{sylv}_{0}$ is a generator of
$(I^{\operatorname{sat}}/I)_{\delta}$. Therefore,
$\\{\operatorname{sylv}_{\mu}\\}_{x^{\mu}\in R_{\nu}}$ yields an $A$-basis of
$(I^{\operatorname{sat}}/I)_{\delta-\nu}$.

In the rest of the paper, we provide three applications to illustrate the
practical interest of toric Sylvester forms in elimination theory. The first
application deals with elimination matrices. An important question in
elimination theory is the study of matrices $\mathbb{M}$ with entries in $A$
such that i) their rank drops when specialized in $k$ to a system having
solutions in $X_{\Sigma}$ and ii) when specialized in $k$ to a system having
finitely many solutions, their corank coincides with this number of solutions.
The first property can be related to resultant theory whilst the second one
has practical implications in solving $0$-dimensional polynomial systems; see
[CDS97, GKZ94] for i) and [BT22, EM99] for ii). In this paper, we introduce a
new family of elimination matrices by adding to a classical Macaulay-block
matrix in some degree $\alpha\in\operatorname{Cl}(X_{\Sigma})$, a block-matrix
built from the toric Sylvester forms of degree $\alpha$; see Definition 5.1.
We call these matrices hybrid elimination matrices and prove their main
properties in Theorem 5.1. Compared with the more classical Macaulay matrices,
this new family yields more compact matrices that can still be used for
solving 0-dimensional polynomial systems. In addition, we also prove that the
construction of hybrid elimination matrices can be extended to polynomial
systems with more than $n+1$ polynomials, providing this way new matrices that
can be used to solve overdetermined 0-dimensional polynomial systems; see
Theorem 5.3.

Our second application concerns the computation of sparse resultants. A
classical result in elimination theory is that the sparse resultant can be
computed as the determinant of certain graded components of the Koszul complex
built from the considered polynomial system; see [GKZ94]. Generalizing a
construction of Cattani, Dickenstein and Sturmfels in [CDS97, §2] using the
so-called toric Jacobian, we modify the usual Koszul complex by incorporating
the Sylvester forms in its last differential and prove that the determinant of
some suitable graded parts of this new complex is equal to the sparse
resultant, up to a nonzero multiplicative constant in $k$; see Theorem 6.1.
This result yields new compact formulas for computing the sparse resultant as
a determinant of a complex.

Our third application deals with the _computation of toric residues_. The
toric residue of the generic polynomial system (1.2) was defined by Cox in
[Cox96]. It is a map that sends any polynomial in $(C/I)_{\delta}$ to the
fraction field $K(A)$ of $A$. The computation of this residue map by means of
determinants has been an active research topic with many contributions,
including [Jou97, DK05, CCD97]. In this paper, using toric Sylvester forms we
construct matrices whose determinants are used to compute the residue of a
product of two forms $PQ$, where $P\in C_{\nu}$, $Q\in C_{\delta-\nu}$ and
$\nu\in\operatorname{Cl}(X_{\Sigma})$. This formula can be seen as an
extension of a similar formula proved by Jouanolou in the case
$X_{\Sigma}=\mathbb{P}^{n}$ [Jou97, Proposition 3.10.27]. It yields more
compact matrices in comparison with the formula proved by D’Andrea and Khetan
in [DK05, Theorem 5.1] for computing the toric residue of a form of degree
$\delta$.

The paper is organised as follows. In Section 2, we present all the tools of
toric geometry that are needed in the rest of the paper. In particular, we
prove the existence of decompositions of forms in a projective toric varieties
$X_{\Sigma}$ satisfying to a certain positivity property. In Section 3, we
show that the claimed duality property holds outside a region
$\Gamma\subset\operatorname{Cl}(X_{\Sigma})$ which depends on the supports of
the local cohomology modules of the Cox ring. In Section 4, we define
Sylvester forms and show that they give an $A$-basis of
$(I^{\operatorname{sat}}/I)_{\delta-\nu}$ for certain degrees
$\nu\in\operatorname{Cl}(X_{\Sigma})$. In Section 5, we introduce hybrid
elimination matrices when $X_{\Sigma}$ is a smooth projective toric variety;
we prove that, under mild conditions, the computation of the local cohomology
modules is not necessary and we extend these results to overdetermined
systems. In Section 6, we prove that the determinant of certain graded parts
of a modified Koszul complex in a region
$\Gamma_{\operatorname{Res}}\subset\operatorname{Cl}(X_{\Sigma})$ is equal to
the sparse resultant, up to a nonzero multiplicative constant in $k$. Finally,
in Section 7 we prove a new formula for computing the toric residue of a
product of two forms.

##  2 Preliminaries on toric geometry

In this section, we set our notation and briefly review some material we will
use from toric geometry; we refer the unfamiliar reader to the book by Cox,
Little and Schenck [CLS12] which offers a detailed treatment of this topic. At
the end of the section, we also prove a decomposition result that we will use
in order to introduce toric Sylvester forms later on.

#### Toric varieties.

Let $k$ be an algebraically closed field and let $M$ be a lattice of rank $n$.
We denote by $N=\operatorname{Hom}(M,\mathbb{Z})$ the dual of $M$, by
$\mathbb{T}_{N}=N\otimes k^{\times}$ the algebraic torus associated to $N$ and
we set $M_{\mathbb{R}}=M\otimes\mathbb{R}$, which are two vector spaces over
the real numbers. Let $\mathcal{A}=\\{m_{1},\dots,m_{s}\\}\subset M$ be a
finite set of lattice points and consider its convex hull
$\Delta=\operatorname{conv}(\mathcal{A})\subset M_{\mathbb{R}}$. The
projective toric variety $X_{\Delta}$ can be defined as the algebraic closure
of the image of the map

$\Phi_{\mathcal{A}}:\mathbb{T}_{N}\xrightarrow{}\mathbb{P}^{s-1}_{k}\quad
t:=(t_{1},\ldots,t_{n})\xrightarrow[]{}(t^{m_{1}}:\dots:t^{m_{s}}).$

This variety is called toric because the group action of $\mathbb{T}_{N}$ on
itself extends to $X_{\Delta}$ with good geometric properties. Another
definition of $X_{\Delta}$, more intrinsic, can be stated from the normal fan
$\Sigma\subset N$ of $\Delta$ so that this variety is also denoted by
$X_{\Sigma}$; see [Cox95].

######  Example 2.1.

If $\Delta$ is a product of simplices of the form
$\Delta_{n_{j}}=\\{t\in\mathbb{R}^{n_{j}}\,:\,t_{k}\geq
0,\,\sum_{k=0}^{n_{j}}t_{k}\leq 1\\}$ for $j=1,\dots,s$, then
$X_{\Sigma}=\mathbb{P}^{n_{1}}\times\dots\times\mathbb{P}^{n_{s}}$.

The geometric properties of $X_{\Sigma}$ are deeply connected with the
combinatorial properties of the fan $\Sigma$. Thus, $X_{\Sigma}$ is a smooth
variety if and only if $\Sigma$ is smooth, which means that the minimal
generators of all cones $\sigma\in\Sigma$ are part of a basis of $N$.
Similarly, $X_{\Sigma}$ is a complete variety if and only if $\Sigma$ is
complete, i.e. if $N_{\mathbb{R}}=\cup_{\sigma\in\Sigma}\sigma$; see [CLS12,
Theorem 3.1.19].

We denote by $\Sigma(r)$ the set of $r$-dimensional cones of $\Sigma$, which
are also called rays when $r=1$. We assume that the generators of the rays
$u_{\rho}\in N$ for $\rho\in\Sigma(1)$ are primitive and span the vector space
$N_{\mathbb{R}}$; by [CLS12, Corollary 3.3.10], this condition is equivalent
to the toric variety $X_{\Sigma}$ having no torus factors. Moreover, as
$\Delta$ is a bounded polytope, its normal fan $\Sigma$ is complete and its
cones are strongly convex. Under these assumptions, $\Sigma(1)$ contains at
least $n+1$ rays.

Let $\operatorname{Cl}(X_{\Sigma})$ be the class group of $X_{\Sigma}$. There
is a short exact sequence

$0\xrightarrow[]{}M\xrightarrow[]{\mathbf{F}}\mathbb{Z}^{\Sigma(1)}\xrightarrow[]{\pi}\operatorname{Cl}(X_{\Sigma})\xrightarrow[]{}0,$
(2.1)

where $\mathbf{F}$ is an $n\times(n+r)$ matrix whose rows are the generators
of the rays in $\Sigma(1)$ and $\pi$ is chosen accordingly to be a cokernel
matrix; see [CLS12, Theorem 4.1.3]. A Cartier divisor $D$ is nef if and only
if $D$ is generated by global sections [CLS12, Theorem 6.3.12], and ample if
and only if the normal fan of its polytope is $\Sigma$ [CLS12, Proposition
7.2.3]. We notice that if $X_{\Sigma}$ is a complete projective toric variety
then there exists at least one ample divisor; see [Har77, Chapter 2, Theorem
7.10]. Because of that, _from now on we assume that $X_{\Sigma}$ is
projective_.

#### The Cox ring and a positivity property.

The homogeneous coordinate ring of a toric variety $X_{\Sigma}$, also known as
Cox ring, is the ring $R=k[x_{\rho},\,\rho\in\Sigma(1)]$ which is
$\operatorname{Cl}(X_{\Sigma})$-graded by the map $\pi$ in (2.1):
$R=\oplus_{\alpha\in\operatorname{Cl}(X_{\Sigma})}R_{\alpha}$, with
$R_{\alpha}=H^{0}(X_{\Sigma},\mathcal{O}_{\Sigma}(D))$ where $D$ is a torus-
invariant Weil divisor such that $[D]=\alpha$ and $\mathcal{O}_{\Sigma}$ is
the structural sheaf of $X_{\Sigma}$; see [Cox95]. We will use the following
notation for the variables of the Cox ring: being given $\sigma\in\Sigma(n)$ a
maximal smooth cone, we will denote by $x_{1},\dots,x_{n}$ the variables
associated to the rays $\rho\in\sigma(1)$ and by $z_{1},\dots,z_{r}$ the
remaining variables. According to the choice of $\sigma$, one can always write
a matrix of the map $\pi$ in (2.1) under the form

$\pi=\begin{pmatrix}\mathcal{P}&\operatorname{Id}_{r}\end{pmatrix},$ (2.2)

$\mathcal{P}$ being a block matrix $(\mathcal{P}_{i,j})_{1\leq i\leq r,1\leq
j\leq n}$, whose rows correspond to the relations between $u_{\rho}$ for
$\rho\notin\sigma$ and the basis given by $\sigma$.

######  Definition 2.1.

We say that $X_{\Sigma}$ has the positivity property with respect to $\sigma$
if there exists a smooth cone $\sigma\in\Sigma(n)$ such that a matrix of the
map $\pi$ can be written as in (2.2) with the additional condition that
$\mathcal{P}_{i,j}\geq 0$ for $i=1,\dots,r$ and $j=1,\dots,n$.

The above positivity property can be understood as saying that the vector
$-u_{\rho}$ belongs to $\sigma$ for all $\rho\notin\sigma(1)$, as each row of
$\pi$ corresponds to the identity
$u_{\rho_{j}}+\sum_{\rho_{i}\in\sigma(1)}\mathcal{P}_{i,j}u_{\rho_{i}}=0.$ A
first observation is that not all smooth toric varieties satisfy this
property, as shown in the following example.

######  Example 2.2.

Let $\Sigma$ be the complete smooth fan in $N_{\mathbb{R}}=\mathbb{R}^{2}$
with the following rays:

$\rho_{1}=(1,0)\>\rho_{2}=(0,1)\>\rho_{3}=(-1,1)\>\rho_{4}=(-1,0)\>\rho_{5}=(-1,-1)\>\rho_{6}=(0,-1).$

It is straightforward to check that for every maximal $\sigma\in\Sigma(2)$,
there is $\rho\notin\sigma(1)$ such that $-u_{\rho}\notin\sigma$.

On the other hand, most of the toric varieties that are of interest for our
applications do satisfy the positivity property.

######  Lemma 2.1.

Any product of toric varieties having the positivity property with respect to
some $\sigma$ has the positivity property with respect to some other
$\sigma^{\prime}$.

###### Proof.

The cones of the fans in these varieties must be a union of cones of each
factor. If one considers $\sigma^{\prime}$ to be the union of all the cones
with respect to which the factors have the positivity property, it must have
this property as well. ∎

######  Example 2.3.

The projective space $\mathbb{P}^{n}$ satisfies the positivity property as the
map $\pi$ can be written as $\pi=(1\cdots 1)$. Therefore, any product of
projective spaces has the positivity property by Lemma 2.1. Another classical
family of smooth toric varieties are Hirzebruch surfaces
$\mathcal{H}_{r}\subset\mathbb{R}^{2}$: for each $r\in\mathbb{Z}_{>0}$, it is
the variety corresponding to the fan $\Sigma_{r}$ with rays

$\rho_{1}=(1,0)\>\rho_{2}=(0,1)\>\rho_{3}=(-1,-r)\>\rho_{4}=(0,-1).$

Hirzebruch surfaces are smooth and have the positivity property with respect
to $\sigma=(\rho_{1},\rho_{2})$ as $\pi$ can be written as

$\pi=\begin{pmatrix}1&r&1&0\\\ 0&1&0&1\end{pmatrix}.$

#### Generic sparse homogeneous polynomial systems.

Let $\Delta_{0},\dots,\Delta_{n}$ be rational polytopes in $M_{\mathbb{R}}$.
Let $\Sigma$ be the normal fan of the Minkowski sum
$\Delta=\sum_{i=0}^{n}\Delta_{i}$ and $X_{\Sigma}$ be the corresponding
projective toric variety. Suppose that $X_{\Sigma}$ has the positivity
property with respect to some cone $\sigma\in\Sigma(n)$. The polytopes
$\Delta_{i}$ can be seen as elements
$a_{i}=(a_{i,j})\in\mathbb{Z}^{\Sigma(1)}$ using the following facet
presentation:

$\Delta_{i}=\\{m\in M_{\mathbb{R}}\,:\,\langle m,u_{j}\rangle\geq-
a_{i,j},\,\rho_{j}\in\Sigma(1)\\},\,i=0,\ldots,n.$ (2.3)

These presentations relate each of the polytopes $\Delta_{i}$ to Weil divisors
that can be written as $\sum_{j}a_{i,j}D_{j}$ where $D_{j}$ is the torus
invariant divisor associated with the ray $\rho_{j}$. Using 2.1, we see that
two polytopes that map to the same class in $\operatorname{Cl}(X_{\Sigma})$
are translations of each other. For each class
$\alpha_{i}\in\operatorname{Cl}(X_{\Sigma})$, we choose this presentation so
that the vertex associated to the cone $\sigma$ is $0\in M$. In particular,
this implies that $a_{i,j}=0$ for $\rho_{j}\in\sigma(1)$.

Let $\alpha_{0},\dots,\alpha_{n}$ be nef classes in
$\operatorname{Cl}(X_{\Sigma})$ associated to $\Delta_{0},\dots,\Delta_{n}$,
and $R_{\alpha_{0}},\dots,R_{\alpha_{n}}$ be the corresponding graded
components in the Cox ring, respectively. These graded components are finite
$k$-vector spaces and have a monomial basis given by
$x^{\mu}:=x_{1}^{\mu_{1}}\cdots x_{n}^{\mu_{n}}z_{1}^{\mu_{n+1}}\cdots
z_{r}^{\mu_{r}}\in R$. Let $A=k[c_{i,\mu}\,:\,x^{\mu}\in
R_{\alpha_{i}},\,i=0,\dots,n]$ and $C=A[x_{1},\dots,x_{n},z_{1},\dots,z_{r}]$.
A generic homogeneous sparse polynomial system of degree
$\alpha_{0},\ldots,\alpha_{n}$ is the system obtained from the polynomials

$F_{i}=\sum_{x^{\mu}\in R_{\alpha_{i}}}c_{i,\mu}x^{\mu}\in
C=A[x_{1},\dots,x_{n},z_{1},\dots,z_{r}],\ \ i=0,\ldots,n.$ (2.4)

The ring $C$ can be interpreted as the Cox ring of the toric variety
$X_{\Sigma}\times_{k}\operatorname{Spec}(A)$ over generic coefficients and its
graded components are given by $C_{\alpha}=R_{\alpha}\otimes_{k}A$.

If the system is dehomogenized by setting $z_{1}=\dots=z_{r}=1$, the Newton
polytopes $F_{i}$ is $\Delta_{i}$ for $i=0,\dots,n$. Conversely, the
polynomials $F_{0},\dots,F_{n}$ can be defined as the homogenization of the
system of polynomials $\tilde{F}_{0}=\dots=\tilde{F}_{n}=0$ with supports in
the subsets $\mathcal{A}_{i}=\Delta_{i}\cap M$ for $i=0,\dots,n$:

$\tilde{F}_{i}=\sum_{m\in\mathcal{A}_{i}}c_{i,m}x^{m}\in\tilde{C}=A[x_{1},\dots,x_{n}]\xrightarrow[]{}{F}_{i}=\sum_{m\in\mathcal{A}_{i}}c_{i,m}x^{\mathbf{F}m+a_{i}}\in{C}=A[x_{1},\dots,x_{n},z_{1},\dots,z_{r}]$

where $\mathbf{F}$ and $a_{i}$ appear in (2.1) and (2.3), respectively. By
homogenizing the monomials associated to the lattice points in
$\mathcal{A}_{i}$, we can choose a monomial basis of $R_{\alpha_{i}}$ using
$\mu=\mathbf{F}m+a_{i}$; see [BT22, Section 2.2] for more details about
homogenization and dehomogenization of sparse polynomial systems.

#### Torsion and local cohomology.

From the fan $\Sigma$ of a toric variety $X_{\Sigma}$, the irrelevant ideal
$\mathfrak{b}$ of its homogeneous coordinate ring
$C=A[x_{\rho},\,\rho\in\Sigma(1)]$ is defined as

$\mathfrak{b}=(x^{\overline{\sigma}}\textrm{ such that
}\sigma\in\Sigma(n)),\textrm{ where
}x^{\overline{\sigma}}=\prod_{\rho\notin\sigma(1)}x_{\rho}.$

The $\mathfrak{b}$-torsion of a graded $C$-module $S$ is classically defined
as

$\Gamma_{\mathfrak{b}}(S)=\\{a\in
S\,:\,\mathfrak{b}^{k}·a=0,\,k\in\mathbb{N}\\}$

and the local cohomology modules $H^{i}_{\mathfrak{b}}(S)$ are then the
derived functors of $S\xrightarrow[]{}\Gamma_{\mathfrak{b}}(S)$. When the
module $S$ is a quotient ring $B=C/I$ for $I=\langle F_{0},\dots,F_{n}\rangle$
the ideal defined by (2.4), the $0$-th local cohomology is
$H_{\mathfrak{b}}^{0}(B)=I^{\operatorname{sat}}/I$ where
$I^{\operatorname{sat}}$ denotes the saturation of the ideal $I$ with respect
to the irrelevant ideal of $C$, i.e.
$I^{\operatorname{sat}}:=(I:\mathfrak{b}^{\infty})=\\{p\in C\,:\,\exists
k\in\mathbb{Z}\quad\mathfrak{b}^{k}p\subset I\\}$.

Local cohomology modules are strongly related to sheaf cohomology modules.
More precisely, let $S$ be a finitely generated
$\operatorname{Cl}(X_{\Sigma})$-graded $R$-module with associated coherent
sheaf $\mathcal{S}$ in $X_{\Sigma}$ and
$\alpha\in\operatorname{Cl}(X_{\Sigma})$. If $p\geq 2$, then

$H_{\mathfrak{b}}^{p}(S)_{\alpha}\simeq
H^{p-1}(X_{\Sigma},\mathcal{S}(\alpha)).$ (2.5)

Furthermore, the following exact sequence holds:

$0\xrightarrow{}H_{\mathfrak{b}}^{0}(S)_{\alpha}\xrightarrow[]{}S_{\alpha}\xrightarrow{}H^{0}(X_{\Sigma},\mathcal{S}(\alpha))\xrightarrow[]{}H_{\mathfrak{b}}^{1}(S)_{\alpha}\xrightarrow[]{}0;$

see [CLS12, Theorem 9.5.7] for proofs. If $S=R$, then
$R_{\alpha}=H^{0}(X_{\Sigma},\mathcal{O}_{\Sigma}(\alpha))$ and therefore

$H^{0}_{\mathfrak{b}}(R)=H^{1}_{\mathfrak{b}}(R)=0,$ (2.6)

which implies that $H^{i}_{\mathfrak{b}}(C)=0$ for $i=0,1$.

######  Notation 2.1.

For the sake of simplicity in the notation, for any Cartier divisor $D$ and
any integer $p\geq 0$, we will write $H^{p}(X_{\Sigma},\alpha)$ in place of
$H^{p}(X_{\Sigma},\mathcal{O}_{\Sigma}(D))$, where
$\alpha=[D]\in\operatorname{Cl}(X_{\Sigma})$.

The theorems of Batyrev-Borisov [BB11, Theorem 2.5] and Demazure [Dem70,
Corollary 1] are the main tools that we will rely on in order to analyze the
vanishing of sheaf cohomology modules of toric varieties. We recall that, if
$\alpha\in\operatorname{Cl}(X_{\Sigma})$ is a nef class, then

  * •

$H^{p}(X_{\Sigma},\alpha)\simeq 0$ for $p>0$ (Demazure),

  * •

$H^{p}(X_{\Sigma},-\alpha)\simeq 0$ if $p\neq\dim\Delta_{\alpha}$ (Batyrev-
Borisov),

  * •

$H^{p}(X,-\alpha)\simeq\oplus_{m\in\operatorname{Relint}(\Delta_{\alpha})\cap
M}k\chi^{-m}$ if $p=\dim\Delta_{\alpha}$ (Batyrev-Borisov).

Another important result we will use is the toric version of Serre duality:
for any $\alpha\in\operatorname{Cl}(X_{\Sigma})$ and any integer $p\geq 0$,

$H^{p}(X_{\Sigma},\alpha)\cong H^{n-p}(X_{\Sigma},-K_{X}-\alpha)^{\vee},$

where $K_{X}$ the anticanonical class in $\operatorname{Cl}(X_{\Sigma})$; see
[CLS12, Theorem 9.2.10] for a proof.

#### Hilbert functions and the Grothendieck-Serre formula.

Let $X_{\Sigma}$ be a smooth projective toric variety, $R$ be its Cox ring and
$S$ a graded $R$-module. The Hilbert function of $S$ is defined as

$\displaystyle\operatorname{HF}(S,-):\operatorname{Cl}(X_{\Sigma})$
$\displaystyle\rightarrow$ $\displaystyle\mathbb{Z}_{\geq 0}$
$\displaystyle\alpha$ $\displaystyle\mapsto$
$\displaystyle\operatorname{HF}(S,\alpha):=\dim_{k}(S_{\alpha}).$

For $\alpha\gg 0$ (component-wise), this function becomes a (multivariate)
polynomial which is called the Hilbert polynomial and is denoted by
$\operatorname{HP}(S,\alpha)$; see [MS03, Lemma 2.8].

######  Remark 2.1.

If $S=R/J$ with $J$ an ideal defining a $0$-dimensional subscheme in
$X_{\Sigma}$, then the Hilbert polynomial of $S$ is a constant which is equal
to the number of points counted with multiplicity.

An important relation between the Hilbert function, the Hilbert polynomial and
local cohomology modules is given by the Grothendieck-Serre formula; see
[MS03, Proposition 2.14]: for any $\alpha\in\operatorname{Cl}(X_{\Sigma})$,

$\operatorname{HF}(S,\alpha)=\operatorname{HP}(S,\alpha)+\sum_{i=0}^{d}(-1)^{i}\dim_{k}H_{\mathfrak{b}}^{i}(S)_{\alpha}.$
(2.7)

#### Existence of some decompositions.

To conclude this preliminary section, we prove the existence of certain
decompositions of the polynomials $F_{i}$, $i=0,\dots,n$, that we will use in
Section 4 in order to define toric Sylvester forms. For sake of clarity, we
state the following theorem over $k$ and with polynomials in $R$, but this
result extends naturally to generic homogeneous polynomials, as defined in
(2.4).

Let $J$ be an ideal of $R$ generate by homogeneous polynomials
$f_{0},\ldots,f_{n}$ of degree $\alpha_{0},\ldots,\alpha_{n}$, respectively.

######  Theorem 2.1.

Let $X_{\Sigma}$ be a projective toric variety over $k$ with the positivity
property with respect to $\sigma$. Let $\nu\in\operatorname{Cl}(X_{\Sigma})$
be a nef class and $\Delta_{\nu}$ be the corresponding polytope, written as in
(2.3), satisfying $0\leq\nu_{j}<\min_{i}a_{i,j}$ for
$\rho_{j}\notin\sigma(1)$. Then, the two following properties hold:

  * •

$R_{\nu}=(R/J)_{\nu}$.

  * •

For every $x^{\mu}\in R_{\nu}$ and $f_{i}\in R_{\alpha_{i}}$ and
$i=0,\dots,n$, there exists a decomposition of the form

$f_{i}=z_{1}^{\mu_{n+1}+1}\cdots
z_{r}^{\mu_{n+r}+1}f^{\mu}_{i,0}+x_{1}^{\mu_{1}+1}f^{\mu}_{i,1}+\dots+x_{n}^{\mu_{n}+1}f^{\mu}_{i,n}$

where the $f_{i,j}^{\mu}$, $i,j=0,\dots,n$, are homogeneous polynomials in
$R$.

###### Proof.

The graded quotient map $R_{\nu}\xrightarrow[]{}(R/J)_{\nu}$ is surjective.
Using the degree constraint, its kernel must be zero, giving the first
property. On the other hand, we have to prove that for
$x^{\mu}=x_{1}^{\mu_{1}}\dots x_{n}^{\mu_{n}}z_{1}^{\mu_{n+1}}\dots
z_{r}^{\mu_{n+r}}\in R_{\nu}$ (recall; $\mu=\mathbf{F}m+\nu$ for
$m\in\mathcal{A}_{i}$), every monomial $x^{\mathbf{F}m+a_{i}}\in R_{i}$ that
is not divided by $z_{1}^{\mu_{n+1}+1}\cdots z_{r}^{\mu_{n+r}+1}$ is divided
by some of the $x_{1}^{\mu_{1}+1},\dots,x_{n}^{\mu_{n}+1}$. Using the toric
homogenization, the fact that the first does not happen implies that:

$\langle u_{n+j},m\rangle+a_{i,j}\leq\mu_{n+j}\quad j=1,\dots,r.$

Considering any of these $j$ and using that
$\nu_{j}=\mu_{n+j}+\sum_{k=1}^{n}\mathcal{P}_{k,j}\mu_{k}$, through the map
$\pi$ defined in Definition 2.1, we get:

$\langle u_{n+j},m\rangle+\min a_{i,j}\leq\mu_{n+j}\implies\langle
u_{n+j},m\rangle+\nu_{j}<\mu_{n+j}\implies\\\ \langle
u_{n+j},m\rangle+\sum_{k=1}^{n}\mathcal{P}_{k,j}\mu_{k}<0\implies\sum_{k=1}^{n}\mathcal{P}_{k,j}(\mu_{k}-\langle
u_{k},m\rangle)<0$

As $X_{\Sigma}$ has the positivity property, then there must be
$k\in\\{1,\dots,n\\}$ such that $\mu_{k}-\langle u_{k},m\rangle<0$. This
implies that $x_{k}^{\mu_{k}+1}$ divides $x^{\mathbf{F}m+a_{i}}$ as the
coefficient of $x_{k}$ in this monomial is $\mathbf{F}u_{k}+a_{i,k}=\langle
u_{k},m\rangle$. ∎

This theorem extends trivially to a decomposition of the generic homogeneous
sparse polynomials $F_{i}$ over $X_{\Sigma}\times_{k}A$, where $F_{i,j}^{\mu}$
are homogeneous polynomials in $C$.

######  Corollary 2.1.

If $\Delta_{i}$ is $n$-dimensional for all $i=0,\dots,n$, then the previous
theorem holds for $\nu=0$.

###### Proof.

If $\Delta_{i}$ is $n$-dimensional, then $a_{i,j}>0$ for $j>n$. Therefore,
$0<\min_{i}a_{i,j}$ for $\rho_{j}\notin\sigma(1)$, implying the result. ∎

##  3 A duality theorem

In this section, some graded components of the saturation of the ideal
generated by $n+1$ generic homogeneous sparse polynomials over a projective
toric variety $X_{\Sigma}$ of dimension $n$, are analyzed via a duality
property. We assume that $X_{\Sigma}$ has the positivity property with respect
to $\sigma\in\Sigma(n)$ and we take again the notation of (2.4):
$F_{0},\dots,F_{n}$ are the generic homogeneous polynomials of degree
$\alpha_{0},\dots,\alpha_{n}$, respectively; they are of the form

$F_{i}=\sum_{x^{\mu}\in C_{i}}c_{i,\mu}x^{\mu}\in
C=A[x_{1},\dots,x_{n},z_{1},\dots,z_{r}].$ (3.1)

As a preliminary result, we show that $F_{0},\dots,F_{n}$ form a regular
sequence outside $V(\mathfrak{b})\subset\operatorname{Spec}(C)$.

######  Lemma 3.1.

The homogeneous generic polynomials $F_{0},\dots,F_{n}$ define a regular
sequence in the localization ring $C_{\sigma}:=C_{x^{\overline{\sigma}}}$ for
any $\sigma\in\Sigma(n)$.

###### Proof.

We claim that $F_{0}$ is a nonzero divisor in $C$. This follows as a corollary
of Dedekind-Mertens Lemma [BJ14, Corollary 2.8], which says that $F$ is a
nonzero divisor in $A[x_{1},\dots,x_{n}]$ if its content ideal is a nonzero
divisor in $A$. The content ideal is generated by the coefficients $c_{0,\mu}$
for $x^{\mu}\in C_{0}$ and they are all nonzero divisors. Therefore, $F_{0}$
is a nonzero divisor also in $C_{\sigma}$ for all $\sigma\in\Sigma(n)$.

Now, as $\Sigma$ always refines the normal fan of $\Delta_{i}$, we can always
find a vertex $a_{\sigma}\in\mathcal{A}_{i}$ corresponding to the maximal cone
$\sigma\in\Sigma(n)$. Let $c_{i,\sigma}$ be the coefficient associated to this
vertex. Then, similarly to [BCN22, Lemma 3.2], for any
$t\in\\{1,\ldots,n-1\\}$ there is a isomorphism of algebras

$B^{t}_{\sigma}=\big{(}A[x_{1},\dots,x_{n},z_{1},\dots,z_{r}]/\langle
F_{0},\dots,F_{t}\rangle\big{)}_{\sigma}\xrightarrow{\sim}(A_{\sigma}^{t}[x_{1},\dots,x_{n},z_{1},\dots,z_{r}])_{\sigma}$

where $A_{\sigma}^{t}=k[c_{i,\mu}\quad c_{i,\mu}\neq c_{i,\sigma}\quad 0\leq
i\leq t]$ and which maps $c_{i,\sigma}x^{a_{\sigma}}$ to
$F_{i}-c_{i,\sigma}x^{a_{\sigma}}$ and which leaves invariant the other
variables and coefficients. Applying again Dedekind-Mertens Lemma as above, we
deduce that $F_{t+1}$ is a nonzero divisor in
$(A_{\sigma}^{t}[x_{1},\dots,x_{n},z_{1},\dots,z_{r}])_{\sigma}$, and
therefore in $B^{t}_{\sigma}$. ∎

Next, we consider the two canonical spectral sequences associated with the
Čech-Koszul double complex
$\mathcal{C}_{\mathfrak{b}}^{\bullet}(K_{\bullet}(F))$, where $K_{\bullet}(F)$
denotes the Koszul complex of the sequence of homogeneous polynomials
$F_{0},\ldots,F_{n}$ in $C$. The terms of this Koszul complex are graded free
$C$-modules and we denote their homology modules by $H_{p}$ for simplicity in
the notation. If we start taking homologies horizontally, the second page is:

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The vanishing of the local cohomology modules $H_{\mathfrak{b}}^{i}(H_{j})$
for $i>0$ and $j>0$ follows from the fact that the $F_{i}$’s form a regular
sequence after localization by a generator of $\mathfrak{b}$ by Lemma 3.1. In
addition, we deduce that $H_{p}$ are geometrically supported on
$V(\mathfrak{b})$ for all $p>0$ by a classical property of Koszul complexes,
and hence that $H_{\mathfrak{b}}^{0}(H_{p})=H_{p}$ for all $p>0$.

On the other hand, if we start taking homologies vertically, we obtain the
following first page:

${0}$${0}$${0}$${\cdots}$${0}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$${H_{\mathfrak{b}}^{n-1}(C(-\sum\alpha_{i}))}$${\rightarrow}$${H_{\mathfrak{b}}^{n-1}(\oplus_{k}C(-\sum_{j\neq
k}\alpha_{j}))}$${\rightarrow}$${H_{\mathfrak{b}}^{n-1}(\oplus_{k,k^{\prime}}C(-\sum_{j\neq
k,k^{\prime}}\alpha_{j}))}$${\cdots}$${H_{\mathfrak{b}}^{n-1}(C)}$${H_{\mathfrak{b}}^{n}(C(-\sum\alpha_{i}))}$${\rightarrow}$${H_{\mathfrak{b}}^{n}(\oplus_{k}C(-\sum_{j\neq
k}\alpha_{j}))}$${\rightarrow}$${H_{\mathfrak{b}}^{n}(\oplus_{k,k^{\prime}}C(-\sum_{j\neq
k,k^{\prime}}\alpha_{j}))}$${\cdots}$${H_{\mathfrak{b}}^{n}(C)}$${H_{\mathfrak{b}}^{n+1}(C(-\sum\alpha_{i}))}$${\rightarrow}$${H_{\mathfrak{b}}^{n+1}(\oplus_{k}C(-\sum_{j\neq
k}\alpha_{j}))}$${\rightarrow}$${H_{\mathfrak{b}}^{n+1}(\oplus_{k,k^{\prime}}C(-\sum_{j\neq
k,k^{\prime}}\alpha_{j}))}$${\cdots}$${H_{\mathfrak{b}}^{n+1}(C)}$

using that $K_{j}(F)=\oplus_{|J|=j}C(-\sum_{k\in J}\alpha_{k})$ for
$J\subset\\{0,\dots,n\\}$. We notice that the vanishing of the two first rows
follows from Remark 2.6, and also that the vanishing of
$H_{\mathfrak{b}}^{p}(C)$ for all $p>n+1$ is a consequence of Grothendieck’s
vanishing Theorem [Gro57, Theorem 3.6.5].

######  Notation 3.1.

The support $\operatorname{Supp}S$ of a graded module $S$ is the subset of
$\nu\in\operatorname{Cl}(X_{\Sigma})$ such that $S_{\nu}\neq 0$. We denote by
$\Gamma_{1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
the support of the modules on the main diagonal, expect on the last row, and
by
$\Gamma_{0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
the support of the modules in the diagonal under
$\Gamma_{1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$,
expect on the last row again, i.e

$\Gamma_{i}=\operatorname{Supp}(\oplus_{p=0}^{n}H_{\mathfrak{b}}^{p}(K_{p+i-1}(F)))\quad
i=0,1$

. In addition, we define $\Gamma_{\operatorname{Res}}$ to be the support of
all the cohomology modules that are appearing above the diagonal in the first
page of the second spectral sequence, i.e.
$\Gamma_{\operatorname{Res}}=\cup_{i<j}\operatorname{Supp}H_{\mathfrak{b}}^{i}(K_{j}(F))$
Moreover, from now on, we denote by $\delta$ the Cartier divisor class
$\alpha_{0}+\dots+\alpha_{n}-K_{X}$ where $K_{X}$ denotes the anticanonical
divisor of $X_{\Sigma}$.

The comparison of the two above spectral sequences leads to the following
duality.

######  Theorem 3.1.

Let $X_{\Sigma}$ be a projective toric variety and let
$\nu\in\operatorname{Cl}(X_{\Sigma})$. If
$\delta-\nu\notin\Gamma_{0}\cup\Gamma_{1}$ then

$(I^{\operatorname{sat}}/I)_{\delta-\nu}\simeq\operatorname{Hom}_{A}((C/I)_{\nu},A).$

###### Proof.

From the comparison of the two spectral sequences associated to the double
complex $\mathcal{C}_{\mathfrak{b}}^{\bullet}(K_{\bullet}(F))$, for all
$\delta-\nu\notin\Gamma_{0}\cup\Gamma_{1}$ we get an isomorphism

$(I^{\operatorname{sat}}/I)_{\delta-\nu}\simeq\operatorname{Ker}\left(H_{\mathfrak{b}}^{n+1}(C(-\sum_{i}\alpha_{i}))\xrightarrow[]{}H_{\mathfrak{b}}^{n+1}(\oplus_{j}C(-\sum_{i\neq
j}\alpha_{i}))\right)_{\delta-\nu}.$

Moreover, using toric Serre duality and the relation between sheaf and local
cohomology modules, we obtain

$H_{\mathfrak{b}}^{n+1}(C(-\sum_{i}\alpha_{i}))_{\delta-\nu}\simeq
H^{n}(X_{\Sigma},-\nu-K_{X})\simeq
H^{0}(X_{\Sigma},\nu)^{\vee}\simeq\operatorname{Hom}_{A}(C_{\nu},A).$

By the same argument, we also have
$H_{\mathfrak{b}}^{n+1}(\oplus_{j}C(-\sum_{i\neq
j}\alpha_{i}))_{\delta-\nu}\simeq\operatorname{Hom}_{A}(I_{\nu},A)$. Using the
first isomorphism, we get the duality property. ∎

Theorem 3.1 holds if
$\delta-\nu\in\operatorname{Supp}H_{\mathfrak{b}}^{n+1}(\oplus_{j}C(-\sum_{i\neq
j}\alpha_{i})))$, which is apriori not contained in $\Gamma_{0}$ or
$\Gamma_{1}$, but if it does not belong to this support, then we get the
following important consequence.

######  Corollary 3.1.

Under the assumption of Theorem 3.1, if in addition
$\delta-\nu\notin\operatorname{Supp}H_{\mathfrak{b}}^{n+1}(\oplus_{j}C(-\sum_{i\neq
j}\alpha_{i})))$ then

$(I^{\operatorname{sat}}/I)_{\delta-\nu}\simeq\operatorname{Hom}_{A}(C_{\nu},A).$

In particular, $(I^{\operatorname{sat}}/I)_{\delta-\nu}$ is a free $A$-module
whose rank is equal to $\operatorname{rank}_{A}C_{\nu}$.

We notice that the additional condition in the above corollary happens
precisely when $\nu$ does not reach the generating degrees of $I$, i.e.
$I_{\nu}=0$, equivalently $C_{\nu}=(C/I)_{\nu}$ as in (2.1). We also notice
that the case $\nu=0$, which corresponds to the isomorphism
$(I^{\operatorname{sat}}/I)_{\delta}\simeq A$, was already known; see [BC93,
CCD97].

##  4 Toric Sylvester forms

By Corollary 3.1, some graded components of $I^{\operatorname{sat}}/I$ are
free $A$-modules and hence a natural question is to provide explicit $A$-bases
for them. This is precisely the goal of this section. We will first describe
the graded component $(I^{\operatorname{sat}}/I)_{\delta}$, which essentially
follows from [CCD97], and then introduce Sylvester forms to deal with the
other cases. In what follows, we take again the notation of Section 3, in
particular $X_{\Sigma}$ is assumed to be a projective toric variety that has
the positivity property with respect to a cone $\sigma$.

Following [CCD97], we find a nonzero element of
$(I^{\operatorname{sat}}/I)_{\delta}\simeq A$ as follows. Using Corollary 2.1,
if $\Delta_{0},\dots,\Delta_{n}$ are $n$-dimensional, one can decompose each
polynomial as

$F_{i}=x_{1}{F}_{i,1}+\dots+x_{n}{F}_{i,n}+z_{1}\cdots z_{r}{F}_{i,n+1},$
(4.1)

and consider the determinant

$\operatorname{Sylv}_{0}=\det\begin{pmatrix}F_{i,j}\end{pmatrix}_{0\leq
i,j\leq n}.$

This homogeneous polynomial is called the toric jacobian; we will denote its
class modulo $I$ by $\operatorname{sylv}_{0}$.

######  Proposition 4.1.

The element $\operatorname{Sylv}_{0}$ belongs to
$(I^{\operatorname{sat}})_{\delta}$. Moreover, $\operatorname{sylv}_{0}$ is
independent on the choices of decompositions (4.1) and the choice of $\sigma$
(as long as $X_{\Sigma}$ has the positivity property with respect to
$\sigma$). In addition, $\operatorname{sylv}_{0}$ is a generator of
$(I^{\operatorname{sat}}/I)_{\delta}$, which is a free $A$-module of rank 1.

###### Proof.

The fact that $\operatorname{Sylv}_{0}\in I^{\operatorname{sat}}$ follows from
$x_{i}\operatorname{Sylv}_{0}\in I$ for $i=1,\dots,n$ and $z_{1}\cdots
z_{r}\operatorname{Sylv}_{0}\in I$. The $A$-module
$(I^{\operatorname{sat}}/I)_{\delta}$ is free of rank $1$ by Theorem 3.1 and
the fact that $C_{0}\simeq A$. The fact that $\operatorname{sylv}_{0}$ is
nonzero is a consequence of [CCD97, Theorem 0.2] and the independence of the
choice of the decomposition (4.1) is a consequence of the classical Wiebe’s
lemma; see [Jou95, Proposition 3.8.1.6].

The fact that $\operatorname{Sylv}_{0}\notin I$ and the independence from
$\sigma$ is proved in [CCD97, Theorem 0.2] using toric residues, namely that
$\operatorname{Residue}_{F}(\operatorname{sylv}_{0})=1$, generically. In loc.
cit. the hypothesis that the $\alpha_{i}$ ’s are $\mathbb{Q}$-ample, for
$i=0,\dots,n$, is used in order to derive the decomposition (4.1). In our
context, we already derived such decomposition in Theorem 2.1 so, as claimed
in [CCD97, Remark 2.12, iv], the same property holds in this case.

In order to prove that $\operatorname{sylv}_{0}$ has degree $\delta$, we find
the degree of each entry $(i,j)$ of the matrix defined by the $F_{i,j}$. In
(4.1), we divided a set of monomials of degree $\alpha_{i}$, by a monomial of
degree

$\begin{cases}\pi(e_{j})&\text{if the monomial is }x_{j}\text{ for
}j=1,\dots,n,\\\ \pi(\sum_{k=n+1}e_{j})&\text{if the monomial is }z_{1}\cdots
z_{r},\end{cases}$

where $\\{e_{j}\\}_{j=1}^{n+r}$ is the canonical basis of
$\mathbb{Z}^{\Sigma(1)}$. On the other hand, the anticanonical class $K_{X}$
coincides with the degree of the monomial $x_{1}\cdots x_{n}z_{1}\cdots
z_{r}$, which is equal to $\pi(\sum_{j=1}^{n+r}e_{j})$. Therefore, the degree
of each of the summands that constitutes the determinant is equal to:

$\sum_{i=0}^{n}\big{(}\alpha_{i}-\pi(e_{\sigma(i)})\big{)}=\left(\sum_{i=0}^{n}\alpha_{i}\right)-K_{X}=\delta,$

where $\sigma\in\mathfrak{S}_{n}$ is the element of the symmetric group
corresponding to such summand. ∎

We notice that using Batyrev-Borisov theorem, the Sylvester form
$\operatorname{sylv}_{0}$ is in correspondence with the unique lattice point
in the interior of the polytope $\Delta_{\Sigma}$ associated to the
anticanonical divisor $K_{X}$.

$(I^{\operatorname{sat}}/I)_{\delta}\simeq
H^{n+1}_{\mathfrak{b}}(C(-\sum\alpha_{i}))_{\delta}\simeq
H^{n}(X_{\Sigma},-K_{X})\simeq\oplus_{m\in\operatorname{Relint}(\Delta_{\Sigma})}A\chi^{-m}.$

We have proved that the toric Jacobian $\operatorname{sylv}_{0}$ yields an
$A$-basis of $(I^{\operatorname{sat}}/I)_{\delta}\simeq A$. Our next step is
to construct an $A$-basis of $(I^{\operatorname{sat}}/I)_{\delta-\nu}$ when
this latter is a free $A$-module.

######  Definition 4.1.

Let $X_{\Sigma}$ be a projective toric variety with the positivity property
with respect to $\sigma$. Let $\nu\in\operatorname{Cl}(X_{\Sigma})$ be a nef
class and $\Delta_{\nu}$ be the corresponding polytope, written as in (2.3),
satisfying $0\leq\nu_{j}<\min_{i}a_{i,j}$ for $\rho_{j}\notin\sigma(1)$. Then,
according to Theorem 2.1, being given $x^{\mu}\in R_{\nu}$, each polynomial
$F_{i}$ can be decomposed as

$F_{i}=z_{1}^{\mu_{n+1}+1}\cdots
z_{r}^{\mu_{n+r}+1}F^{\mu}_{i,0}+x_{1}^{\mu_{1}+1}F^{\mu}_{i,1}+\dots+x_{n}^{\mu_{n}+1}F^{\mu}_{i,n}$
(4.2)

and the toric Sylvester form $\operatorname{Sylv}_{\mu}$ is then defined as
the determinant

$\operatorname{Sylv}_{\mu}=\det(F^{\mu}_{i,j})_{0\leq i,j\leq n}.$

The class of $\operatorname{Sylv}_{\mu}$ modulo $I$ is denoted by
$\operatorname{sylv}_{\mu}$.

######  Theorem 4.1.

Let $X_{\Sigma}$ be a projective toric variety. Then, for any
$\nu\in\operatorname{Cl}(X_{\Sigma})$ satisfying the hypotheses of Theorem 2.1
and any pair $x^{\mu},x^{\mu^{\prime}}\in R_{\nu}$:

$x^{\mu^{\prime}}\operatorname{sylv}_{\mu}=\begin{cases}\operatorname{sylv}_{0}&\mu=\mu^{\prime}\\\
0&\text{otherwise.}\end{cases}$

The element $\operatorname{Sylv}_{\mu}$ belongs to
$(I^{\operatorname{sat}})_{\delta-\nu}$. Its class
$\operatorname{sylv}_{\mu}\in(I^{\operatorname{sat}}/I)_{\delta-\nu}$ is a
nonzero element which is independent of the choices of decompositions and of
$\sigma$ (as long as $X_{\Sigma}$ has the positivity property with respect to
$\sigma$). Therefore, $\\{\operatorname{sylv}_{\mu}\\}_{x^{\mu}\in C_{\nu}}$
gives a basis of $(I^{\operatorname{sat}}/I)_{\delta-\nu}$.

###### Proof.

First, the fact that $\operatorname{Sylv}_{\mu}$ has degree $\delta-\nu$
follows by using the same reasonning as the one at the end of Proposition 4.1.
Now, from the decompositions (4.2) and the invariance of the determinant by
column operations, we get

$x_{j}^{\mu_{j}+1}\operatorname{Sylv}_{\mu}=\det\blockarray{ccc}\block{(ccc)}\cdots&x_{j}^{\mu_{j}+1}F_{0,j}\cdots\\\
\cdots\vdots\cdots\\\ \cdots x_{j}^{\mu_{j}+1}F_{0,j}\cdots\\\
=\det\blockarray{ccc}\block{(ccc)}\cdots&F_{0}\cdots\\\ \cdots\vdots\cdots\\\
\cdots F_{n}\cdots\\\ \in I,$

and the same holds for the monomial $z_{1}^{\mu_{n+1}+1}\cdots
z_{r}^{\mu_{n+r}+1}$. This proves that $\operatorname{Sylv}_{\mu}\in
I^{\operatorname{sat}}_{\delta-\nu}$. Suppose that for $x^{\mu}\neq
x^{\mu^{\prime}}\in R_{\nu}$, there exists $j\in\\{1,\dots,n\\}$ such that
$\mu^{\prime}_{j}>\mu_{j}$ and $x_{j}^{\mu_{j}+1}$ divides $x^{\mu^{\prime}}$.
Then,

$x^{\mu^{\prime}}\operatorname{Sylv}_{\mu}=\frac{x^{\mu^{\prime}}}{x_{j}^{\mu_{j}+1}}x_{j}^{\mu_{j}+1}\operatorname{Sylv}_{\mu}\in
I\implies
x^{\mu^{\prime}}\operatorname{sylv}_{\mu}=0\in(I^{\operatorname{sat}}/I)_{\delta-\nu}.$

If this does not happen, then $\mu^{\prime}_{j}\leq\mu_{j}$ for all
$j\in\\{1,\dots,n\\}$. Using the positivity property, this implies that
$\sum_{j=1}^{n}\mathcal{P}_{j,k}\mu^{\prime}_{j}\leq\sum_{j=1}^{n}\mathcal{P}_{j,k}\mu_{j}$
for $k=1,\dots,r$, but if it was an equality, then:

$\nu_{k}=\mu^{\prime}_{n+k}+\sum_{j=1}\mathcal{P}_{j,k}\mu^{\prime}_{j}=\mu_{n+k}+\sum_{j=1}\mathcal{P}_{j,k}\mu_{j}\quad
k=1,\dots,r$

where we used that $x^{\mu}$ and $x^{\mu^{\prime}}$ have the same degree
$\nu$. This would imply that $x^{\mu}=x^{\mu^{\prime}}$, a contradiction.
Otherwise, $\mu^{\prime}_{n+k}>\mu_{n+k}$ for all $k=1,\dots,r$, implying:

$x^{\mu^{\prime}}\operatorname{Sylv}_{\mu}=\frac{x^{\mu^{\prime}}}{z_{1}^{\mu_{n+1}+1}\cdots
z_{r}^{\mu_{n+r}+1}}z_{1}^{\mu_{n+1}+1}\cdots
z_{r}^{\mu_{n+r}+1}\operatorname{Sylv}_{\mu}\in I\implies
x^{\mu^{\prime}}\operatorname{sylv}_{\mu}=0\in(I^{\operatorname{sat}}/I)_{\delta-\nu}.$

On the other hand, we have

$x^{\mu}\operatorname{Sylv}_{\mu}=x_{1}^{\mu_{j}}\cdots
x_{n}^{\mu_{n}}z_{1}^{\mu_{n+1}}z_{r}^{\mu_{n+r}}\det(F_{i,j})=\det(x_{j}^{\mu_{j}}F_{i,j})$

but at the same time, the decomposition

$F_{i}=x_{1}x_{1}^{\mu_{1}}F_{i,1}+x_{2}x_{2}^{\mu_{2}}F_{i,2}+\dots+z_{1}\cdots
z_{r}z_{1}^{\mu_{n+1}}\cdots z_{r}^{\mu_{n+r}}F_{i,n+1}$

gives the Sylvester form $\operatorname{sylv}_{0}$, implying the equality and
that $\operatorname{Sylv}_{\mu}\notin I$. The fact that they form a basis
follows from the duality in Theorem 3.1 and that $\operatorname{sylv}_{0}$ is
a basis of $(I^{\operatorname{sat}}/I)_{\delta}$ ∎

######  Remark 4.1.

We notice that over the field $k$, the duality between Sylvester forms and
monomials in Theorem 4.1 can also be deduced from the global transformation
law in [CCD97, Theorem 0.2]. The approach we developed above allows us to work
over the ring $A$, which is the universal ring of coefficients over $k$.

##  5 Application to toric elimination matrices

Our main motivation for studying the structure of the saturation of an ideal
generated by generic sparse polynomials and for introducing Sylvester forms as
generalizations of the toric Jacobian is for applications in elimination
theory, in particular for solving sparse polynomial systems. Thus, in this
section we introduce and study a new family of matrices whose construction
uses toric Sylvester forms. It yields compact elimination matrices that can be
used for solving 0-dimensional sparse polynomial systems via linear algebra
methods; we refer the reader to [EM99, BT22, Tel20] for a thorough exposition
of such solving methods that we will not discuss in this paper.

In what follows, we will consider a smooth projective toric variety
$X_{\Sigma}$ and a generic sparse polynomial system defined by homogeneous
polynomials $F_{0},\dots,F_{n}$ as defined in (2.4). We need to assume
additionally that $X_{\Sigma}$ is smooth because we will use the Grothendieck-
Serre formula. However, we notice that this assumption is not strongly
restrictive as $X_{\Sigma}$ could be replaced by one of its desingularization
variety. Some desingularization procedures are described in [CLS12, Chapters
10,11]; if this variety has the positivity property with respect to some
$\sigma$, then our approach can be applied by considering the same polytopes
as nef divisors in this desingularization.

The elimination matrices we will consider are universal with respect to the
coefficients of the $F_{i}$’s, so we introduce the following notation to state
and study rigorously their properties under specialization of these
coefficients. Recall that $I$ denotes the ideal in $C$ generated by
$F_{0},\ldots,F_{n}$.

######  Notation 5.1.

Any specialization (i.e. ring morphism) $\rho:A\xrightarrow[]{}k$ induces a
surjective map $C\xrightarrow[]{}R$ where $R=k[x_{\rho}\,:\,\rho\in\Sigma(1)]$
(this map leaves invariant the variables $x_{\rho}$). For all $i=0,\ldots,n$
we define $f_{i}=\rho(F_{i})\in R$ and we denote by $I(f)$ the homogeneous
ideal $(f_{0},\ldots,f_{n})$ of $R$ and set $B(f)=R/I(f)$. Moreover, we also
set $B^{\operatorname{sat}}=C/I^{\operatorname{sat}}$,
$B(f)^{\operatorname{sat}}=R/I(f)^{\operatorname{sat}}$ and
$B^{\operatorname{sat}}=C/I^{\operatorname{sat}}(f)$ (observe that
$I(f)^{\operatorname{sat}}$ and $I^{\operatorname{sat}}(f)$ are in general not
the same ideals). Finally, for any matrix $\mathbb{M}$ with coefficients in
$A$, we denote by $\mathbb{M}(f)$ its specialization by
$\rho:A\xrightarrow[]{}k$. We also consider $\operatorname{Pic}(X_{\Sigma})$
instead of $\operatorname{Cl}(X_{\Sigma})$ as all Weil divisors are Cartier in
a smooth variety; see [CLS12, Proposition 4.2.6].

###  5.1 Hybrid elimination matrices

We begin by describing precisely what we mean by an elimination matrix
$\mathbb{M}$ associated to the polynomials $F_{0},\ldots,F_{n}$. It as a
matrix whose columns are filled with coefficients of some homogeneous forms
that are of the same degree and that belong to the saturated ideal
$I^{\operatorname{sat}}\subset C$. Thus, its entries are polynomials in $A$.
Moreover, it is required that for any specialization map
$\rho:A\xrightarrow[]{}k$ the following two properties hold :

  * i)

The corank of $\mathbb{M}(f)$ is equal to zero if and only if
$f_{0}=\dots=f_{n}=0$ has no solution in $X_{\Sigma}$.

  * ii)

If the number of solutions of $f_{0}=\dots=f_{n}=0$ is finite in $X_{\Sigma}$
and equals $\kappa$, then the corank of $\mathbb{M}(f)$ is $\kappa$.

Some comments are in order about this definition. The first property yields a
certificate of existence of a common root of the $f_{i}$’s. We notice that
this is equivalent to the vanishing of their sparse resultant; we will come
back to this topic in the next section. The second property is mainly required
for solving 0-dimensional polynomial systems by means of linear algebra
techniques based on eigen-computations, as in this approach the common roots
of the $f_{i}$’s are extracted from the cokernel of $\mathbb{M}(f)$. We also
emphasize that requiring the columns of $\mathbb{M}$ to be filled with
homogeneous forms in $I^{\operatorname{sat}}$ is also mainly targeted for
solving 0-dimensional polynomial systems. Indeed, this guarantees that each
common root to the $f_{i}$’s corresponds to a sub-vector space of the cokernel
of $\mathbb{M}(f)$, and then the second property implies that the total
dimension of these spaces is the total number of roots, counted with
multiplicity. From a more geometric point of view, homogeneous forms in
$I^{\operatorname{sat}}$ specialize to hypersurfaces in $X_{\Sigma}$ that
vanish scheme-theoretically at the roots of the $f_{i}$’s in $X_{\Sigma}$; see
[Cox95, Corollary 3.8]. For this reason, in the elimination theory literature
these forms are sometimes called inertia forms of the polynomial
$F_{0},\ldots,F_{n}$.

A very classical family of elimination matrices is obtained by filling columns
with all the multiples of the $F_{i}$’s of a certain degree. These matrices
are usually called Macaulay-type matrices and are widely used for solving
0-dimensional polynomial systems; see [BT22]. To be more precise, these
matrices, that we denote by $\mathbb{M}_{\alpha}$, are presentation matrices
of the $A$-module $B_{\alpha}$, i.e. are the matrices of the maps

$\displaystyle\big{(}\oplus_{i=0}^{n}C(-\alpha_{i}))_{\alpha}$
$\displaystyle\xrightarrow[]{}$ $\displaystyle C_{\alpha}$ (5.1)
$\displaystyle(G_{0},\dots,G_{n})$ $\displaystyle\mapsto$
$\displaystyle\sum_{i=0}^{n}G_{i}F_{i}.$

Of course, some conditions on $\alpha\in\operatorname{Pic}(X_{\Sigma})$ are
required in order to guarantee that $\mathbb{M}_{\alpha}$ is an elimination
matrix; see [EM99] or [Tel20, Chapter 5] for good reviews on the properties of
Macaulay-type matrices. In our setting, we see them as a particular case of
the hybrid construction we now describe.

Applying results we proved in the previous sections, we introduce a new family
of elimination matrices by using toric Sylvester forms. We recall that
Sylvester forms belong to $I^{\operatorname{sat}}$ by Theorem 4.1.

######  Definition 5.1.

Let $\alpha$ be such that
$\left(I^{\operatorname{sat}}/I\right)_{\alpha}\simeq\oplus_{\mu}A$ is a free
$A$-module; e.g. $\alpha\notin\Gamma_{0}\cup\Gamma_{1}$ and
$I_{\delta-\alpha}=0$ as in Corollary 3.1. Consider the map

$\displaystyle\left(\oplus_{i=0}^{n}C(-\alpha_{i})\right)_{\alpha}\oplus\left(I^{\operatorname{sat}}/I\right)_{\alpha}$
$\displaystyle\rightarrow$ $\displaystyle C_{\alpha}$ (5.2)
$\displaystyle(G_{0},\ldots,G_{n},l_{\mu})$ $\displaystyle\mapsto$
$\displaystyle\sum_{i=0}^{n}G_{i}F_{i}+\sum_{x^{\mu}\in
C_{\delta-\alpha}}l_{\mu}\operatorname{Sylv}_{\mu}$

where $l_{\mu}\in A$. Its matrix is called an hybrid elimination matrix and
will be denoted by $\mathbb{H}_{\alpha}$.

The matrices $\mathbb{H}_{\alpha}$ are called _hybrid_ because they are
composed of two blocks, one from the classical Macaulay-type matrices and
another one built from toric Sylvester forms. In particular, we notice that
$\mathbb{M}_{\alpha}=\mathbb{H}_{\alpha}$ if
$(I^{\operatorname{sat}}/I)_{\alpha}=0$, so that the family of matrices
$\mathbb{H}_{\alpha}$ can be seen as an extension of the family of Macaulay-
type matrices $\mathbb{M}_{\alpha}$; from now on we will use the notation
$\mathbb{H}_{\alpha}$ instead of $\mathbb{M}_{\alpha}$. Showing that these
matrices are elimination matrices is the purpose of the next section.

###  5.2 Main properties

In this section, we first prove that the matrices $\mathbb{H}_{\alpha}$
introduced in Definition 5.1 are elimination matrices. Then, we give an
illustrative example and also provide another criterion to construct the
matrices $\mathbb{H}_{\alpha}$ without relying on the computation of the
supports $\Gamma_{0}$ and $\Gamma_{1}$.

First, suppose given a specialization map (see Notation 5.1) and a degree
$\alpha$. From the results of Section 3 and Section 4, and also Definition
5.1, we deduce that the image of the matrix $\mathbb{H}_{\alpha}(f)$ is
$I^{\operatorname{sat}}(f)_{\alpha}$, so that its corank is
$\operatorname{HF}(B^{\operatorname{sat}}(f),\alpha)$. Therefore, a natural
question is to compare this Hilbert function of $B^{\operatorname{sat}}(f)$
with the one of $B(f)^{\operatorname{sat}}$ in degrees for which hybrid
matrices $\mathbb{H}$ are defined; see Definition 5.1.

######  Lemma 5.1.

Let
$\alpha\notin\Gamma_{0}\cup\Gamma_{1}\subset\operatorname{Pic}(X_{\Sigma})$
and suppose given specialized polynomials $f_{0},\ldots,f_{n}$ defining a
0-dimensional subscheme in $X_{\Sigma}$, possibly empty, of $\kappa$ points,
counted with multiplicity. Then,

$\operatorname{HF}(B(f)^{\operatorname{sat}},\alpha)=\operatorname{HF}(B^{\operatorname{sat}}(f),\alpha)=\kappa.$

###### Proof.

This proof goes along the same lines as [BCN22, Lemma 2.7]. First, one
observes that $I(f)\subset I^{\operatorname{sat}}(f)\subset
I(f)^{\operatorname{sat}}$ so that $B(f)^{\operatorname{sat}}$,
$B^{\operatorname{sat}}(f)$ and $B(f)$ have the same Hilbert polynomial, which
is the constant $\kappa$ by our assumption.

Now, $H^{i}_{\mathfrak{b}}(B(f)^{\operatorname{sat}})=0$ for $i=0$ and for all
$i>1$ since $V(I(f))$ is finite. Applying Grothendieck-Serre formula, it
follows that $\operatorname{HF}(B(f)^{\operatorname{sat}},\alpha)=\kappa$ for
all $\alpha$ such that
$H^{1}_{\mathfrak{b}}(B(f)^{\operatorname{sat}})_{\alpha}=0$. Analyzing the
two spectral sequences associated to the Čech-Koszul complex of
$f_{0},\ldots,f_{n}$, we get that the above vanishing holds for all
$\alpha\notin\Gamma_{0}\cup\Gamma_{1}$.

Similarly, Grothendieck-Serre formula and the finiteness of $V(I(f))$ imply
that $\operatorname{HF}(B^{\operatorname{sat}}(f),\alpha)=\kappa$ for all
$\alpha$ such that
$H^{0}_{\mathfrak{b}}(B(f)^{\operatorname{sat}})_{\alpha}=H^{1}_{\mathfrak{b}}(B(f)^{\operatorname{sat}})_{\alpha}=0$.
By [Cha13, Proposition 6.3], the vanishing of these modules can be derived
from the similar vanishing conditions
$H^{0}_{\mathfrak{b}}(B^{\operatorname{sat}})_{\alpha}=H^{1}_{\mathfrak{b}}(B^{\operatorname{sat}})_{\alpha}=0$.
These latter conditions hold for all $\alpha\notin\Gamma_{0}\cup\Gamma_{1}$,
which concludes the proof. ∎

######  Remark 5.1.

As a consequence of the above lemma, we notice that, under its assumptions,
the canonical map from $I^{\operatorname{sat}}_{\alpha}$ to
$I(f)^{\operatorname{sat}}_{\alpha}$, which is induced by a specialization
$\rho$, is surjective, i.e. generators of $I(f)^{\operatorname{sat}}_{\alpha}$
can be computed by means of universal formulas.

######  Theorem 5.1.

Let
$\alpha\notin\Gamma_{0}\cup\Gamma_{1}\subset\operatorname{Pic}(X_{\Sigma})$.
Then, $\mathbb{H}_{\alpha}$ is an elimination matrix, that is:

  * •

$\mathrm{corank}(\mathbb{H}_{\alpha}(f))=0$ if and only if $V(I(f))$ is empty
in $X_{\Sigma}$,

  * •

If $V(I(f))$ is a finite subscheme of degree $\kappa$ in $X_{\Sigma}$, then
$\mathrm{corank}(\mathbb{H}_{\alpha}(f))=\kappa$.

###### Proof.

The proof of the second item follows directly from Lemma 5.1.

For the first item, if $V(I(f))$ is empty, equivalently
$B(f)^{\operatorname{sat}}=0$, then
$\operatorname{HF}(B^{\operatorname{sat}}(f),\alpha)=0$ by Lemma 5.1. If
$V(I(f))\neq\emptyset$, then the $f_{i}$’s have a common solution, say the
point $p\in X_{\Sigma}$ (over $k$) with defining ideal $I_{p}$ (radical and
maximal in $R$). Therefore, since $I^{\operatorname{sat}}(f)\subset
I(f)^{\operatorname{sat}}\subset I_{p}$ and
$\operatorname{HF}(R/I_{p},\beta)=1$ for all
$\beta\in\operatorname{Pic}(X_{\Sigma})$ by the maximality of $I_{p}$, we
deduce that $\operatorname{HF}(R/I^{\operatorname{sat}}(f),\alpha)\neq 0$, for
any $\alpha$. ∎

We emphasize that the above results include the classical Macaulay-type
matrices we mentioned in the previous section.

######  Example 5.1.

Let $M=\mathbb{Z}^{2}$ and $X_{\Sigma}$ be the Hirzebruch surface
$\mathcal{H}_{1}$ described in Example 2.3. Consider the following polytope
presentations:

$\Delta_{i}=\\{m\in\mathbb{Z}^{2}\,:\,\langle m,(1,0)\rangle\geq 0,\>\langle
m,(0,1)\rangle\geq 0,\>\langle m,(-1,-1)\rangle\geq 2,\>\langle
m,(0,-1)\rangle\geq 1\\},\,i=0,1,2.$

$\mathcal{H}_{1}$ has the positivity property with respect to
$\sigma=\langle(1,0),(0,1)\rangle$. The class in
$\operatorname{Pic}(\mathcal{H}_{1})=\mathbb{Z}^{2}$ corresponding to these
polytopes is $\alpha_{i}=(2,1)$ and we write the corresponding generic sparse
homogeneous polynomials as:

$F_{0}=a_{0}z_{1}^{2}z_{2}+a_{1}x_{1}z_{1}z_{2}+a_{2}x_{1}^{2}z_{2}+a_{3}x_{2}z_{1}+a_{4}x_{1}x_{2}\quad\text{resp.
}F_{1},F_{2}\text{ with coefficients }b_{i},c_{i}\quad i=0,\dots,4$

[capbesideposition=right,center,capbesidewidth=10cm]figure
(3,1)(3,2)(4,2)(2,1)

Figure 1: This is the picture of the regions
$\Gamma_{0},\Gamma_{1},\Gamma_{\operatorname{Res}},\Gamma\subset\operatorname{Pic}(X_{\Sigma})=\mathbb{Z}^{2}$
(being the last one defined in Section 6). The blue region corresponds to
$\Gamma_{0}$, the red region corresponds to $\Gamma_{1}$, the green region
corresponds to $\Gamma_{\operatorname{Res}}$ and the brown region corresponds
to $\Gamma$. We marked in orange those $\alpha$ with
$(I^{\operatorname{sat}}/I)_{\alpha}\neq 0$. We derived the local cohomology
of $\mathcal{H}_{1}$ from [Alt+20]; see also [EMS00, Bot11].

We obtain the diagram in the Figure 1 for the supports
$\Gamma_{0},\Gamma_{1},\Gamma_{\operatorname{Res}}$ (also $\Gamma$, which will
be defined in Section 6) and we deduce that some elimination matrices
$\mathbb{H}_{\alpha}$ are obtained for
$\alpha\in\\{(4,2),(3,2),(3,1),(2,1)\\}$. In the cases $\alpha=(4,2)$ and
$\alpha=(3,2)$, we get two Macaulay-type matrices ($\mathbb{M}_{\alpha}$); see
Example 6.1 for more details on their relation with sparse resultant theory.
The two other cases give matrices:

  * •

$\alpha=(3,1)$. This matrix corresponds to $\alpha=\delta$ and in this case,
we are introducing a Sylvester form. This form is $\operatorname{Sylv}_{0}$
and can be computed, as before, by a determinant that we write as:

$\det\blockarray{ccc}\block{(ccc)}a_{1}z_{1}z_{2}+a_{2}x_{1}z_{2}+a_{4}x_{2}&a_{3}z_{1}a_{0}z_{1}\\\
b_{1}z_{1}z_{2}+b_{2}x_{1}z_{2}+b_{4}x_{2}b_{3}z_{1}b_{0}z_{1}\\\
c_{1}z_{1}z_{2}+c_{2}x_{1}z_{2}+c_{4}x_{2}c_{3}z_{1}c_{0}z_{1}\\\
=[130]z_{1}^{3}z_{2}+[230]x_{1}z_{1}^{2}z_{2}+[430]x_{2}z_{1}^{2},$

where
$[ijk]=\scalebox{0.8}{\mbox{$\displaystyle\det\blockarray{ccc}\block{(ccc)}a_{i}&a_{j}a_{k}\\\
b_{i}b_{j}b_{k}\\\ c_{i}c_{j}c_{k}\\\ $}}.$ Therefore, the elimination matrix
$\mathbb{H}_{\alpha}$ is of the form:

$\centering\mathbb{H}_{(3,1)}=\begin{pmatrix}a_{0}&0&b_{0}&0&c_{0}&0&[130]\\\
a_{1}&a_{0}&b_{1}&b_{0}&c_{1}&c_{0}&[230]\\\
a_{2}&a_{1}&b_{2}&b_{1}&c_{2}&c_{1}&0\\\ 0&a_{2}&0&b_{2}&0&c_{2}&0\\\
a_{3}&0&b_{3}&0&c_{3}&0&[430]\\\ a_{4}&a_{3}&b_{4}&b_{3}&c_{4}&c_{3}&0\\\
0&a_{4}&0&b_{4}&0&c_{4}&0\end{pmatrix}.\@add@centering$

This type of matrices for $\alpha=\delta$ were already known from [CDS97] as
the $\Delta_{i}$’s are all equal and ample in $\mathcal{H}_{1}$. However, we
notice that the block of Sylvester forms is more sparse than if we used the
one appearing in that article.

  * •

For $\alpha=(2,1)$, we obtain the following matrix $\mathbb{H}_{\alpha}$ which
is built from two different Sylvester forms:

$\mathbb{H}_{(2,1)}=\begin{pmatrix}a_{0}&b_{0}&c_{0}&\left[013\right]&\left[023\right]\\\
a_{1}&b_{1}&c_{1}&\left[023\right]+[014]&[024]+[123]\\\
a_{2}&b_{2}&c_{2}&[024]&[124]\\\ a_{3}&b_{3}&c_{3}&0&0\\\
a_{4}&b_{4}&c_{4}&0&0\end{pmatrix}$

where the Sylvester forms correspond to the monomial basis $\\{z_{1},x_{1}\\}$
in $C_{\nu}$ for $\nu=(1,0)$. As far as we know, this kind of matrices are
new.

######  Example 5.2.

Let’s change the system of Example 5.1 by setting $\alpha_{2}=(1,1)$. This
implies that the corresponding generic sparse homogeneous polynomial is:

$\displaystyle F_{2}$
$\displaystyle=c_{0}z_{1}z_{2}+c_{1}x_{1}z_{2}+c_{3}x_{2}.$ (5.3)

Notice that, in this case, the Newton polytopes $\Delta_{i}$’s are not scaled
copies of a fixed ample class and $\alpha_{2}$ is not even ample in
$\mathcal{H}_{1}$. Now, $\delta=(2,1)$ and the corresponding Sylvester form
would be

$\det\blockarray{ccc}\block{(ccc)}a_{1}z_{1}z_{2}+a_{2}x_{1}z_{2}+a_{4}x_{2}&a_{3}z_{1}a_{0}z_{1}\\\
b_{1}z_{1}z_{2}+b_{2}x_{1}z_{2}+b_{4}x_{2}b_{3}z_{1}b_{0}z_{1}\\\
c_{1}z_{2}c_{3}c_{0}\\\
=[130]z_{1}^{2}z_{2}+[230]x_{1}z_{1}z_{2}+[430]x_{2}z_{1},$

where
$[ijk]:=\scalebox{0.8}{\mbox{$\displaystyle\det\blockarray{ccc}\block{(ccc)}a_{i}&a_{j}a_{k}\\\
b_{i}b_{j}b_{k}\\\ c_{i}c_{j}c_{k}\\\ $}}$, with the convention that $c_{i}=0$
if this coefficient does not appear in $F_{2}$. The corresponding elimination
matrix is

$\mathbb{H}_{(2,1)}=\begin{pmatrix}a_{0}&b_{0}&c_{0}&0&\left[013\right]\\\
a_{1}&b_{1}&c_{1}&c_{0}&\left[023\right]+[014]\\\ a_{2}&b_{2}&0&c_{1}&0\\\
a_{3}&b_{3}&c_{3}&0&[024]\\\ a_{4}&b_{4}&0&c_{3}&0\end{pmatrix}.$

This example shows that we generalized the results of [CCD97] for
$\alpha=\delta$ with the only restriction that the $\Delta_{i}$ must be
$n$-dimensional.

As illustrated in the above Example 5.1, the construction of the elimination
matrices $\mathbb{H}_{\alpha}$ requires the computation of the support of the
local cohomology module $H^{i}_{\mathfrak{b}}(R)$ over the toric variety
$X_{\Sigma}$; see [Alt+20] for some interesting results, specially when
$\Sigma$ splits or the rank of $\operatorname{Pic}(X_{\Sigma})$ is $2$ or $3$,
and also [EMS00, Bot11]. However, none of these references covers the case of
a general smooth projective variety. The next result provides settings for
which these computations can be avoided.

######  Notation 5.2.

We use the same notation as in Section 3 and the beginning of this section,
writing $\alpha_{i}\in\operatorname{Pic}(X_{\Sigma})$ for the classes
associated to the homogeneous polynomial system, $K_{X}$ for the anticanonical
divisor, $\delta=\alpha_{0}+\dots+\alpha_{n}-K_{X}$ and
$\nu,\alpha\in\operatorname{Pic}(X_{\Sigma})$ as elements in the class group;
$K_{j}(F)$ for the modules involved in the Koszul complex and
$J\subset\\{0,\dots,n\\}$ for a subset of indices.

######  Theorem 5.2.

Assume that all the polytopes $\Delta_{i}$ are $n$-dimensional and that
$\alpha\in\operatorname{Pic}(X_{\Sigma})$ satisfies one of the two following
properties:

  * i)

$\alpha=\delta+\nu$ with $\nu\in\operatorname{Pic}(X_{\Sigma})$ a nef class,

  * ii)

$\alpha=\delta-\nu$ for $\nu\in\operatorname{Pic}(X_{\Sigma})$ a nef class
satisfying the hypotheses of Theorem 2.1, and $\alpha_{i}-\nu$ is nef for all
$i=0,\dots,n$.

Then, $\mathbb{H}_{\alpha}$ is an elimination matrix. Moreover, it is purely
of Macaulay-type if and only if $\alpha$ satisfies i).

###### Proof.

We begin with case i). If $\alpha=\delta+\nu$, we have

$H_{\mathfrak{b}}^{i}(K_{j}(F))_{\delta+\nu}\simeq
H_{\mathfrak{b}}^{i}(\oplus_{|J|=j}C(-\sum_{i\in
J}\alpha_{i}))_{\delta+\nu}\simeq\oplus_{|J|=j}H_{\mathfrak{b}}^{i}(C)_{\delta+\nu-\sum_{i\in
J}\alpha_{i}},$

where we use that the local cohomology functors are exact. Using (2.5), for
$i\geq 2$ and $J\subset\\{0,\dots,n\\}$, we get:

$H_{\mathfrak{b}}^{i}(C)_{\delta+\nu-\sum_{i\in J}\alpha_{i}}\simeq
H^{i-1}(X_{\Sigma},\sum_{i\notin J}\alpha_{i}-K_{X}+\nu)\simeq
H^{n-i+1}(X_{\Sigma},-\sum_{i\notin J}\alpha_{i}-\nu),$

which is zero by Batyrev-Borisov Theorem. The vanishing of
$H_{\mathfrak{b}}^{i}(C)$ for $i=0,1$ follows from (2.6). From here, we check
that
$H_{\mathfrak{b}}^{0}(B)_{\alpha}\simeq(I^{\operatorname{sat}}/I)_{\alpha}=0$
and $H_{\mathfrak{b}}^{1}(B)_{\alpha}=0$, as expected.

Now, we turn to the case ii). We have:

$H_{\mathfrak{b}}^{i}(K_{j}(F))_{\delta-\nu}\simeq
H_{\mathfrak{b}}^{i}(\oplus_{|J|=j}C(-\sum_{i\in
J}\alpha_{i}))_{\delta-\nu}\simeq\oplus_{|J|=j}H_{\mathfrak{b}}^{i}(C)_{\delta-\nu-\sum_{i\in
J}\alpha_{i}}.$

For $i>1$, using Serre duality, we have:

$H^{i}_{\mathfrak{b}}(C)_{\delta-\nu-\sum_{j\in J}\alpha_{j}}\simeq
H^{i-1}(X_{\Sigma},\sum_{j\notin J}\alpha_{j}-K_{X}-\nu)\simeq
H^{n-i+1}(X_{\Sigma},\nu-\sum_{j\notin J}\alpha_{j}).$

As we supposed that $\alpha_{i}-\nu$ is nef, if $J\neq\\{0,\dots,n\\}$,
$\sum_{j\notin J}\alpha_{j}-\nu$ is also nef and we can apply Batyrev-Borisov
Theorem. Moreover, if $J=\\{0,\dots,n\\}$, $H^{n-i+1}(X_{\Sigma},\nu)$
vanishes by Demazure Theorem, unless $i=n+1$. In such case, we have
$H^{n+1}_{\mathfrak{b}}(K_{n+1})_{\delta-\nu}\simeq C_{\nu}$. The claimed
result follows using the fact that $\delta-\nu\notin\Gamma_{0}\cup\Gamma_{1}$
and $(I^{\operatorname{sat}}/I)_{\delta-\nu}$ has a basis of toric Sylvester
forms. ∎

######  Corollary 5.1.

If $\Delta_{i}$ are $n$-dimensional, $\mathbb{H}_{\delta}$ is an elimination
matrix.

###### Proof.

If the $\alpha_{i}$ are full-dimensional, $\nu=0$ is nef and satisfies
$0\leq\nu_{j}<\min\alpha_{ij}$, and the $\alpha_{i}$ are nef so the previous
result holds for $\delta$. ∎

######  Example 5.3.

Refering to Example 5.1, we see that many of the interesting matrices appear
using this setting. The matrix of Macaulay-type with $\alpha=(4,2)$
corresponds to the case i), while the matrix with $\alpha=(3,2)$ does not
belong to either of the two cases as $\nu=(0,1)$ is not a nef divisor;
similarly to the Dixon elimination matrices in [BCN22, Example 5.6]. The
matrices with $\alpha=(3,1),(2,1)$ correspond to case ii) as $\nu=(0,0),(1,0)$
are nef divisors and so are $\alpha_{i}-\nu=(2,1),(1,1)$ for $i=0,1,2$.

###  5.3 Overdetermined sparse polynomial systems

Overdetermined polynomial systems, i.e. systems defined by $r+1$ equations
with $r>n$, appear very often in applications. In this section, we extend the
construction of hybrid elimination matrices to this setting. For that purpose,
the key property we will prove is that all Sylvester forms built from subsets
of $n+1$ polynomial equations in an overdetermined system give a generating
set of suitable graded components of inertia forms. This is the content of
Theorem 5.4.

######  Notation 5.3.

Let $X_{\Sigma}$ be a smooth projective toric variety. In this subsection
$F_{0},\dots,F_{r}$ are generic homogeneous sparse polynomials corresponding
to the nef classes $\alpha_{0},\dots,\alpha_{r}$ and $I$ is their
corresponding ideal, $B=C/I$ their quotient ring. For each subset
$T\subset\\{0,\dots,r\\}$ of cardinality $n+1$, we set $I_{T}=(F_{i}\,:\,i\in
T)$, $B_{T}=C/I_{T}$ and $\delta_{T}=\sum_{i\in T}\alpha_{i}-K_{X}$. We denote
by $\operatorname{Sylv}_{\mu,T}$ the Sylvester forms that can be formed from
$\\{F_{i}\\}_{i\in T}$; see Section 4. We also denote by $K_{\bullet}(F)$ the
Koszul complex of $F_{0},\ldots,F_{r}$ and by $K_{T,\bullet}(F)$ the Koszul
complex built from the generators of $I_{T}$.

######  Theorem 5.3.

Using the previous notation, suppose that there is a subset
$S\subset\\{0,\dots,r\\}$ of cardinality $n+1$ and a nef class
$\nu\in\operatorname{Pic}(X_{\Sigma})$ satisfying the hypotheses of Theorem
2.1 such that

$\forall i\in S\quad j\notin S\quad\alpha_{i}-\alpha_{j}\text{ nef}\ \textrm{
and }\ \forall i\in S\quad\alpha_{i}-\nu\text{ is nef}.$

Then, the set of Sylvester forms

$\\{\operatorname{sylv}_{\mu,T}\,:\,T\subset\\{0,\dots,r\\}\textrm{ such that
}|T|=n+1\textrm{ and }x^{\mu}\in C_{\delta_{T}-\delta_{S}+\nu}\\}$

yields a generating set of the $A$-module
$(I^{\operatorname{sat}}/I)_{\delta_{S}-\nu}$.

###### Proof.

This proof can be seen an generalization of a similar result that is proved in
the case $X_{\Sigma}=\mathbb{P}^{n}$ in [BCP23, Chapter 3, Proposition 3.23].

First, we use Serre duality and Batyrev-Borisov Theorem in order to compute
the local cohomology modules
$H_{\mathfrak{b}}^{i}(K_{j}(F))_{\delta_{S}-\nu}$, for $i,j=0,\dots,n+1$,
similarly to Theorem 5.2. Namely, for $i\geq 2$

$H_{\mathfrak{b}}^{i}(K_{j}(F))_{\delta_{S}-\nu}\simeq\oplus_{|T|=j}H_{\mathfrak{b}}^{i}(C(-\sum_{k\in
T}\alpha_{k}))_{\delta_{S}-\nu}\simeq H^{n+1-i}(X_{\Sigma},\sum_{k\in
T}\alpha_{k}-\sum_{k^{\prime}\in S}\alpha_{k^{\prime}}+\nu).$

The elements in $S\cap T$ cancel each other, and the rest of elements
$k^{\prime}\in S$ can be either (i) paired up with $\alpha_{k}$ for $k\in T$
satisfying that $\alpha_{i}-\alpha_{j}$ is nef, (ii) be paired up with $\nu$
satisfying that $\alpha_{k^{\prime}}-\nu$ is nef or, (iii) they are nef
themselves. Therefore, the previous cohomology module is of the form
$H^{n+1-i}(X_{\Sigma},-\alpha)$ with $\alpha$ a sum of nef divisors, and we
can apply Batyrev-Borisov Theorem to deduce:

$H_{\mathfrak{b}}^{i}(K_{j}(F))_{\delta_{S}-\nu}\simeq\begin{cases}\oplus_{|T|=n+1}C^{\vee}_{\sum_{j\in
T}\alpha_{j}-\sum_{i\in S}\alpha_{i}+\nu}&i,j=n+1\\\
0&\text{otherwise}.\end{cases}$

As a consequence, from the comparison of the two spectral sequences that are
considered in Theorem 3.1, we obtain the following transgression map, which is
an isomorphism of graded modules:

$\tau:H_{n+1}(K_{\bullet}(F),H^{n}_{\mathfrak{b}}(C))_{\delta_{S}-\nu}\xrightarrow{\sim}H^{0}_{\mathfrak{b}}(B)_{\delta_{S}-\nu}.$

For any $T\subset\\{0,\dots,r\\}$, let $\tau_{T}$ be the corresponding
transgression map for $K_{T,\bullet}(F)$ and $B_{T}$. For each of these Koszul
complexes, we have a canonical morphism of complexes
$K_{T,\bullet}(F)\xrightarrow[]{}K_{\bullet}(F)$ that all together induce the
morphism of complexes:

$L_{\bullet}(F)=\bigoplus_{|T|=n+1}K_{T,\bullet}(F)\xrightarrow[]{}K_{\bullet}(F).$

It follows that there is a commutative diagram:

${\oplus_{|T|=n+1}H_{n+1}(K_{T,\bullet}(F),H_{\mathfrak{b}}^{n+1}(C))_{\delta_{S}-\nu}}$${H_{n+1}(K_{\bullet}(F),H_{\mathfrak{b}}^{n+1}(C))_{\delta_{S}-\nu}}$${\oplus_{|T|=n+1}H^{0}_{\mathfrak{b}}(B_{T})_{\delta_{S}-\nu}}$${H_{\mathfrak{b}}^{0}(B)_{\delta_{S}-\nu}}$$\scriptstyle{\oplus_{T}\tau_{T}}$$\scriptstyle{\tau}$
(5.4)

As the two vertical arrows are isomorphisms, in order to show that the bottom
arrow is surjective, it is enough to show that the top arrow is surjective.
For that purpose, observe that $L_{n+1}=K_{n+1}$ by construction and also

$\oplus_{|T|=n+1}H_{n+1}(K_{T,\bullet}(F),H_{\mathfrak{b}}^{n+1}(C))_{\delta_{S}-\nu}=\ker(H_{\mathfrak{b}}^{n+1}(L_{n+1}(F))\xrightarrow[]{}H_{\mathfrak{b}}^{n+1}(L_{n}(F)))_{\delta_{S}-\nu}.$

However, by the same argument as before
$H_{\mathfrak{b}}^{n+1}(L_{n}(F)))_{\delta_{S}-\nu}=0$, so

$\oplus_{|T|=n+1}H_{n+1}(K_{T,\bullet}(F),H_{\mathfrak{b}}^{n+1}(C))_{\delta_{S}-\nu}\simeq\oplus_{|T|=n+1}C^{\vee}_{\sum_{j\in
T}\alpha_{j}-\sum_{i\in S}\alpha_{i}+\nu}.$

which is generated by the Sylvester forms at each of these degrees. On the
other hand,

$H_{n+1}(K_{\bullet}(F),H_{\mathfrak{b}}^{n+1}(C))_{\delta_{S}-\nu}\simeq\\\
\ker(H_{\mathfrak{b}}^{n+1}(K_{n+1})_{\delta_{S}-\nu}\xrightarrow[]{}H_{\mathfrak{b}}^{n+1}(K_{n})_{\delta_{S}-\nu})/\operatorname{Im}(H_{\mathfrak{b}}^{n+1}(K_{n+2})_{\delta_{S}-\nu}\xrightarrow[]{}H_{\mathfrak{b}}^{n+1}(K_{n+1})_{\delta_{S}-\nu}).$

As above, $H_{\mathfrak{b}}^{n+1}(K_{n})_{\delta_{S}-\nu}=0$ and:

$H_{n+1}(K_{\bullet}(F),H_{\mathfrak{b}}^{n+1}(C))_{\delta_{S}-\nu}\simeq\\\
H_{\mathfrak{b}}^{n+1}(K_{n+1})_{\delta_{S}-\nu}/\operatorname{Im}(H_{\mathfrak{b}}^{n+1}(K_{n+2})_{\delta_{S}-\nu}\xrightarrow[]{}H_{\mathfrak{b}}^{n+1}(K_{n+1})_{\delta_{S}-\nu}).$

This implies that the top map in the diagram (5.4) is surjective, as we wanted
to prove. It follows that the basis of Sylvester forms of
$\oplus_{|T|=n+1}H_{\mathfrak{b}}^{0}(B_{T})_{\delta_{S}-\nu}$ is a set of
generators of
$H_{\mathfrak{b}}^{0}(B)_{\delta_{S}-\nu}=(I^{\operatorname{sat}}/I)_{\delta_{S}-\nu}$.
∎

######  Definition 5.2.

Let $\nu\in\operatorname{Pic}(X_{\Sigma})$ satisfying the assumptions of
Theorem 5.3. We will denote by $\mathbb{H}_{\alpha}$ the matrix of the
following map:

$\displaystyle\left(\oplus_{i=0}^{n}C(-\alpha_{i})\right)_{\alpha}\bigoplus_{\begin{subarray}{c}T\subset\\{0,\dots,r\\}\\\
|T|=n+1\end{subarray}}\left(I_{T}^{\operatorname{sat}}/I_{T}\right)_{\alpha}$
$\displaystyle\rightarrow$ $\displaystyle C_{\alpha}$ (5.5)
$\displaystyle(G_{0},\ldots,G_{n},l_{\mu,T})$ $\displaystyle\mapsto$
$\displaystyle\sum_{i=0}^{n}G_{i}F_{i}+\sum_{\begin{subarray}{c}T\subset\\{0,\dots,r\\}\\\
|T|=n+1\end{subarray}}\sum_{x^{\mu}\in
C_{\delta_{T}-\alpha}}l_{\mu,T}\operatorname{Sylv}_{\mu,T}$

where $l_{\mu,T}\in A$.

######  Theorem 5.4.

If $\alpha=\delta_{S}-\nu$ for $S\subset\\{0,\dots,r\\}$ and
$\nu\in\operatorname{Pic}(X_{\Sigma})$ a nef class satisfying the hypotheses
of Theorem 5.3, then $\mathbb{H}_{\alpha}$ is an elimination matrix.

###### Proof.

The proof follows the same lines as the one given in Theorem 5.1 for $r=n$. ∎

######  Example 5.4.

Taking again the notation of Example 2.3, we add another polynomial with
degree $\alpha_{3}=(2,1)$ in $\mathcal{H}_{1}$ and write it in homogeneous
coordinates as

$F_{3}=d_{0}z_{1}^{2}z_{2}+d_{1}x_{1}z_{1}z_{2}+d_{2}x_{1}^{2}z_{2}+d_{3}x_{2}z_{1}+d_{4}x_{1}x_{2}.$

Following Theorem 5.4, the matrix $\mathbb{H}_{\alpha}$ for
$\alpha=\delta_{S}$ is

$\centering\setcounter{MaxMatrixCols}{12}\begin{pmatrix}a_{0}&0&b_{0}&0&c_{0}&0&d_{0}&0&[130]_{abc}&[130]_{abd}&[130]_{acd}&[130]_{bcd}\\\
a_{1}&a_{0}&b_{1}&b_{0}&c_{1}&c_{0}&d_{1}&d_{0}&[230]_{abc}&[230]_{abd}&[230]_{acd}&[230]_{bcd}\\\
a_{2}&a_{1}&b_{2}&b_{1}&c_{2}&c_{1}&d_{2}&d_{1}&0&0&0&0\\\
0&a_{2}&0&b_{2}&0&c_{2}&0&d_{2}&0&0&0&0\\\
a_{3}&0&b_{3}&0&c_{3}&0&d_{3}&0&[430]_{abc}&[430]_{abd}&[430]_{acd}&[430]_{bcd}\\\
a_{4}&a_{3}&b_{4}&b_{3}&c_{4}&c_{3}&d_{4}&d_{3}&0&0&0&0\\\
0&a_{4}&0&b_{4}&0&c_{4}&0&d_{4}&0&0&0&0\end{pmatrix}\@add@centering$

where
$[ijk]_{abc}=\scalebox{0.8}{\mbox{$\displaystyle\blockarray{ccc}\block{(ccc)}a_{i}&a_{j}a_{k}\\\
b_{i}b_{j}b_{k}\\\ c_{i}c_{j}c_{k}\\\ $}}$, and
$[ijk]_{abd},[ijk]_{acd},[ijk]_{bcd}$ defined accordingly. It is an
elimination matrix for the overdetermined polynomial system defined by
$F_{0},F_{1},F_{2}$ and $F_{3}$.

##  6 Sylvester forms and sparse resultants

Resultants are central tools in elimination theory and there is a huge
literature on various methods to compute them. A classical result is that the
sparse resultant can be computed as the determinant of certain graded
components of the Koszul complex built from the considered polynomial system;
see for instance [GKZ94, DD00, WZ92, Ben+21]. In this section, we show that
Sylvester forms can be incorporated in the usual Koszul complex and obtain
this way new expressions for the sparse resultant as the determinant of a
complex. This extends results in [CDS97, §2] by providing more compact
formulas.

In what follows, we assume that $X_{\Sigma}$ is a smooth projective toric
variety and we take again the notation of Section 3. Thus, $F_{0},\dots,F_{n}$
are generic homogeneous sparse polynomial system as defined by (2.4). Let
$\mathcal{A}_{i}$ be the supports of these polynomial system for
$i=0,\dots,n$, respectively, which can be seen as the lattice points in
$\Delta_{i}\subset M$; see Section 2.

The space of coefficients of the $F_{i}$’s has a natural structure of multi-
projective space, as the equations $F_{i}=0$ are not modified after
multiplication by a nonzero scalar. We write it as
$\prod_{i=0}^{n}\mathbb{P}^{\mathcal{A}_{i}}$, where
$\mathbb{P}^{\mathcal{A}_{i}}$ denotes the projective space associated to the
coefficients of the polynomial $F_{i}$. Let
$Z(F)=\\{x\times(\ldots,c_{i,\mu},\ldots)\in
X_{\Sigma}\times\prod_{i=0}^{n}\mathbb{P}^{\mathcal{A}_{i}}\quad{F}_{0}=\cdots={F}_{n}=0\\}$
be the incidence variety of $F_{0},\ldots,F_{n}$. Consider the canonical
projection onto the second factor

$\pi:X_{\Sigma}\times\prod_{i=0}^{n}\mathbb{P}^{\mathcal{A}_{i}}\xrightarrow[]{}\prod_{i=0}^{n}\mathbb{P}^{\mathcal{A}_{i}}.$

The image of $Z(F)$ via $\pi$ is an irreducible hypersurface and the sparse
resultant, denoted as $\operatorname{Res}_{\mathcal{A}}$, is a primitive
polynomial in the universal ring of coefficients $A$ defining the direct image
$\pi_{*}(Z(\mathbf{F}))$, up to a sign in $k$, if it is of codimension one,
otherwise the sparse resultant is set to 1.

The Koszul complex $K_{\bullet}(F)$ of the sequence of polynomials
$F_{0},\ldots,F_{n}$ is of the form

$K_{\bullet}(F):K_{n+1}=C(-\sum\alpha_{i})\xrightarrow[]{}\dots\xrightarrow[]{}K_{2}=\oplus_{k,k^{\prime}}C(-\alpha_{k}-\alpha_{k^{\prime}})\xrightarrow[]{}K_{1}=\oplus_{k}C(-\alpha_{k})\rightarrow
C.$

It is a graded complex of free graded $A$-modules. We already used it several
times previously. Gelfand, Kapranov and Zelevinsky proved that the determinant
of some of its graded components is equal to the sparse resultant, up to a
nonzero scalar in $k$. More precisely, for all
$\alpha\notin\Gamma_{\operatorname{Res}}\subset\operatorname{Pic}(X_{\Sigma})$
the strand $K_{\bullet}(F)_{\alpha}$ is an acyclic complex of free $A$-modules
and $H_{0}(K_{\bullet}(F)_{\alpha})=B_{\alpha}$. Moreover, if in addition
$(I^{\operatorname{sat}}/I)_{\alpha}=0$ then $\det(K_{\bullet}(F)_{\alpha})$
equals the sparse resultant $\operatorname{Res}_{\mathcal{A}}$ up to a nonzero
scalar; see [GKZ94, Chapter 3, Theorem 4.2] for proofs. We notice that the map
on the far right of the complex $K_{\bullet}(F)_{\alpha}$ is nothing but the
Macaulay-type map (5.1), whose matrix is an elimination matrix of the form
$\mathbb{M}_{\alpha}$. In what follows, we will extend this approach by using
Sylvester forms.

In order to incorporate Sylvester forms in the above construction we proceed
as follows: we consider a graded strand $K_{\bullet}(F)_{\alpha}$ of the
Koszul complex, such that $(I^{\operatorname{sat}}/I)_{\alpha}$ is a nonzero
free $A$-module, and we define a new complex, denoted
${K^{\operatorname{sat}}_{\bullet}(F)}_{\alpha}$, by adding Sylvester forms to
the map on the far right, i.e.

$K_{\bullet}^{\operatorname{sat}}(F)_{\alpha}=C(-\sum\alpha_{i})_{\alpha}\xrightarrow[]{d_{n}}\dots\xrightarrow[]{}\oplus_{k,k^{\prime}}C(-\alpha_{k}-\alpha_{k^{\prime}})_{\alpha}\xrightarrow[]{d_{1}}\oplus_{k}C(-\alpha_{k})_{\alpha}\oplus(I^{\operatorname{sat}}/I)_{\alpha}\xrightarrow[]{}C_{\alpha}$

which is a graded complex of free $A$-module, that we call the saturated
Koszul complex. From its definition, it satisfies
$H_{0}(K_{\bullet}^{\operatorname{sat}}(F)_{\alpha})=(B^{\operatorname{sat}})_{\alpha}$.
Moreover, the map on the far right is precisely the map we used to define
hybrid elimination matrices $\mathbb{H}_{\alpha}$. Notice also that if
$(I^{\operatorname{sat}}/I)_{\alpha}=0$, then we recover the usual Koszul
complex.

The following result generalizes [GKZ94, Chapter 3, Theorem 4.2], as well as
[CDS97, Theorem 2.2] where a formula for the sparse resultant as the
determinant of a complex incorporating the toric Jacobian, which only applies
for $\alpha=\delta$ and the polytopes $\Delta_{i}$ being scaled copies of a
given polytope.

######  Theorem 6.1.

Let $\alpha\notin\Gamma_{\operatorname{Res}}$, then
$K_{\bullet}^{\operatorname{sat}}(F)_{\alpha}$ is an acyclic complex of free
$A$-modules. If $\alpha=\delta-\nu$ as in Theorem 5.2 ii), then
$\det(K^{\operatorname{sat}}_{\bullet}(F)_{\alpha})$ is equal to
$\operatorname{Res}_{\mathcal{A}}$ up to a nonzero multiplicative constant in
$k$.

###### Proof.

The acyclicicity of $K_{\bullet}^{\operatorname{sat}}(F)_{\alpha}$ follows
from the the same property on the Koszul complex $K_{\bullet}(F)_{\alpha}$
(observe that the image of $d_{1}$ does not map to
$(I^{\operatorname{sat}}/I)_{\alpha}$) and the fact that
$(I^{\operatorname{sat}}/I)_{\alpha}$ is a free $A$-module.

The acyclicity property, together with the fact that
$H_{0}(K_{\bullet}^{\operatorname{sat}}(F)_{\alpha})=(B^{\operatorname{sat}})_{\alpha}$,
imply that $\det(K_{\bullet}^{\operatorname{sat}}(F)_{\alpha})$ and
$\operatorname{Res}_{\mathcal{A}}$ are two polynomials in $A$ that vanish in
the same specializations in $k$. As a consequence of the Projective
Nullstellenstaz, we only have to compare their degrees in order to check that
they are the same polynomial, up to multiplication by a nonzero constant in
$k$; see [Har77]. As proved in [GKZ94, Section 3, Theorem 14], the determinant
of a complex of vector spaces
$V_{\bullet}:V_{n+1}\xrightarrow[]{}\dots\xrightarrow[]{}V_{1}\xrightarrow[]{}V_{0}$
satisfies the equality

$\det(V_{\bullet})=\bigotimes_{i}\bigwedge^{\operatorname{rk}(F_{i})}V_{i}^{(-1)^{i}}.$

This result implies that the degree of the determinant can be calculated as an
alternate sum. We know that
$\det(K^{\operatorname{sat}}_{\bullet}(F)_{\alpha})=\operatorname{Res}_{\mathcal{A}}$
(up to multiplication by a nonzero constant) if
$(I^{\operatorname{sat}}/I)_{\alpha}=0$ and that for $\alpha\gg 0$ (component-
wise), we have $\operatorname{HF}(C,\alpha)=\operatorname{HP}(C,\alpha)$.
Therefore, for $\alpha\gg 0$,

$\deg(\operatorname{Res}_{\mathcal{A}})=\deg\det(K^{\operatorname{sat}}_{\bullet}(F)_{\alpha})=\sum_{J\subset\\{0,\dots,n\\}}(-1)^{|J|}\operatorname{HF}(C,\alpha-\sum_{j\in
J}\alpha_{i})=\sum_{J\subset\\{0,\dots,n\\}}(-1)^{|J|}\operatorname{HP}(C,\alpha-\sum_{j\in
J}\alpha_{i}).$

This alternate sum yields a polynomial whose degree coincides with the degree
of the resultant. Therefore, for $\alpha=\delta-\nu$ as in the statement, we
have
$(I^{\operatorname{sat}}/I)_{\delta-\nu}=\operatorname{Hom}_{A}(C_{\nu},A)\neq
0$ and we can check that the difference of degrees between the previous
alternate sum and the degree of the resultant is compensated by
$(I^{\operatorname{sat}}/I)_{\delta-\nu}$ as follows:

$\deg\det(K_{\bullet}(F)_{\delta-\nu})-\deg(\operatorname{Res}_{\mathcal{A}})=\sum_{J\subset\\{0,\dots,n\\}}(-1)^{|J|}\operatorname{HF}(C,\delta-\nu-\sum_{j\in
J}\alpha_{j})-\operatorname{HP}(C,\delta-\nu-\sum_{j\in J}\alpha_{j}).$

Using Grothendieck-Serre formula (2.7), we get that this coincides with the
quantity

$\sum_{J\subset\\{0,\dots,n\\}}(-1)^{|J|}\sum_{i=0}^{n+1}(-1)^{i}\dim_{k}H^{i}_{\mathfrak{b}}(C)_{\delta-\nu-\sum_{j\in
J}\alpha_{j}}.$

Applying Theorem 5.2 ii), all the elements in this sum vanish except the term
$H^{n+1}_{\mathfrak{b}}(C)_{-K_{X}-\nu}$, which is counted with the sign
$(-1)^{2(n+1)}=1$. In particular, using Serre duality,
$H^{n+1}_{\mathfrak{b}}(C)_{-K_{X}-\nu}\cong C_{\nu}$ which is dual to
$(I^{\operatorname{sat}}/I)_{\delta-\nu}$, so this difference is compensated
in the complex $K^{\operatorname{sat}}_{\bullet}(F)_{\delta-\nu}$, which
concludes the proof. ∎

In particular, if the polytopes $\Delta_{i}$ are $n$-dimensional for
$i=0,\dots,n$, we can consider $\nu=0$ and we recover the results in [CDS97],
with the slight improvement that we do not need the $\alpha_{i}$ to be of the
form $k_{i}\beta$ for $k_{i}>0$ and $\beta$ an ample class.

######  Remark 6.1.

The relation between $(I^{\operatorname{sat}}/I)_{\delta-\nu}$ and
$\operatorname{HP}(C,\nu)$ in the previous result can be seen as an instance
of multivariate Ehrhart reciprocity; see [Bec02, Theorem 2]. This result shows
that if $\operatorname{HP}(C,\nu)$ is the multivariate Hilbert polynomial in
$X_{\Sigma}$ corresponding to the number of lattice points in $\Delta_{\nu}$
and $\operatorname{HP}^{\circ}(C,\nu)$ is another Hilbert polynomial
associated to the number of lattice points in the interior of $\Delta_{\nu}$,
then $\operatorname{HP}(C,\nu)=(-1)^{n}\operatorname{HP}^{\circ}(C,\nu)$.

From the above result, we can also identify cases where the matrices
$\mathbb{H}_{\alpha}$ are square matrices, and therefore their determinant (in
the usual sense of the determinant of a matrix) is equal to the sparse
resultant, up to a nonzero multiplicative constant.

######  Corollary 6.1.

Let
$\Gamma\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\operatorname{Supp}\oplus_{k,k^{\prime}}C(-\alpha_{k}-\alpha_{k^{\prime}})$.
For $\alpha\notin\Gamma$, we have
$\det(\mathbb{H}_{\alpha})=\operatorname{Res}_{\mathcal{A}}$, up to a nonzero
multiplicative constant.

###### Proof.

If $\alpha\notin\Gamma$, then the complex $K_{\bullet}(F)_{\alpha}$ has only
two terms and therefore
$\det(K_{\bullet}(F)_{\alpha})=\det(\mathbb{H}_{\alpha})$. ∎

######  Remark 6.2.

Computing the determinant of a complex can be done using some techniques such
as Cayley determinants; see [GKZ94, Appendix A], but it is not very practical.
However, Theorem 6.1 yields new expressions of the sparse resultant as a ratio
of two determinants if
$\alpha\notin\operatorname{Supp}\oplus_{k,l,m}C(-\alpha_{k}-\alpha_{l}-\alpha_{m})$;
see [CDS97, Corollary 2.4] for a combinatorial characterization of such case.

For Macaulay-type formulas of the form $\mathbb{M}_{\alpha}$, the Canny-Emiris
formula gives a possible way to choose a nonzero minor; see [CE93] for the
formula and [DJS22] for a proof and the non-vanishing of the minor using
tropical deformations. It is an open problem to see whether the conditions on
the proof of the Canny-Emiris formula [DJS22] coincide with the Cayley
determinant for such choice of a minor. In the case of hybrid elimination
matrices $\mathbb{H}_{\alpha}$, the Canny-Emiris formula has only been
explored in for $n=2$ and $\alpha=\delta$; see [DE01].

######  Example 6.1.

Let’s consider the four matrices provided in Example 5.1, which correspond to
$\alpha\in\\{(4,2),(3,2),(3,1),(2,1)\\}$. The last three are square matrices
while the first one is not. We have drawn the region $\Gamma$ in brown in
Figure 1, in order to indicate the elements that provide a square matrix, as
well as $\Gamma_{\operatorname{Res}}$, in green, for the acyclicity of the
complex. For the Macaulay-type matrices, we can combinatorially describe a
maximal minor of $\mathbb{M}_{(4,2)}$ using the Canny-Emiris formula; see
[CE93, DJS22]. The matrix $\mathbb{H}_{(3,2)}$ is square,

$\mathbb{H}_{(3,2)}=\begin{pmatrix}a_{0}&0&0&b_{0}&0&0&c_{0}&0&0\\\
a_{1}&a_{0}&0&b_{1}&b_{0}&0&c_{1}&c_{0}&0\\\
a_{2}&a_{1}&a_{0}&b_{2}&b_{1}&b_{0}&c_{2}&c_{1}&c_{0}\\\
0&a_{2}&a_{1}&0&b_{2}&b_{1}&0&c_{2}&c_{1}\\\ 0&0&a_{2}&0&0&b_{2}&0&0&c_{2}\\\
a_{3}&0&0&b_{3}&0&0&c_{3}&0&0\\\ a_{4}&a_{3}&0&b_{4}&b_{3}&0&c_{4}&c_{3}&0\\\
0&a_{4}&a_{3}&0&b_{4}&b_{3}&0&c_{4}&c_{3}\\\
0&0&a_{4}&0&0&b_{4}&0&0&c_{4}\end{pmatrix},$

and it might be obtained using a greedy approach to the same formula (see
[CP93, CE22]), but as far as we know, there was no certificate of its
existence as a resultant formula until now. The hybrid matrices for
$\alpha=(3,1),(2,1)$ are square, but if they weren’t, a procedure for choosing
a minor is known for $n=2$ and $\alpha=\delta=(3,1)$; see [DE01].

##  7 Toric residue of the product of two forms

Another topic for which Sylvester forms are of interest is the computation of
toric residues. These objects were initially introduced by Cox as a way to
relate the residue of a family of $n+1$ forms to the integral of a certain
form in a toric variety $X_{\Sigma}$; see [Cox96]. Being given
$F_{0},\ldots,F_{n}$ generic homogeneous polynomials as in (2.4), and denoting
by $K(A)$ the quotient field of the universal ring of coefficients $A$, Cox
proved the existence of a residue map

$\operatorname{Residue}_{F}:B_{\delta}\xrightarrow[]{}K(A)$

(recall that $I=(F_{0},\ldots,F_{n})$ and $B=C/I$) which has the following
property: for any specialization $\rho:A\xrightarrow[]{}k$ (see Notation 5.1)
such that the specialized system $f_{0}=\dots=f_{n}=0$ has no solution in
$X_{\Sigma}$, the residue map
$\operatorname{Residue}_{f}:(R/I(f))_{\delta}\xrightarrow{}k$ is an
isomorphism. Cox defined residue maps through trace maps of Čech cohomology,
but it can be characterized through the fact that, if there is no solution in
$X_{\Sigma}$, $\rho(\operatorname{sylv}_{0})$ is sent to $\pm 1\in k$, so
generically $\operatorname{Residue}_{F}(\operatorname{sylv}_{0})=\pm 1$, as we
used in Proposition 4.1. Many authors contributed formulas based on
elimination matrices and resultants to compute residues [KS05, DK05, CCD97,
CDS97] and also used them in other applications such as code theory [Sop07] or
mirror symmetry [BM02]. In particular, in [DK05] an explicit formula for
computing the toric residue of a form of degree $\delta$ as a quotient of two
determinants “à la Macaulay” is proved.

If a form $G$ of degree $\delta$ can be written as a product $G=PQ$, a natural
question is to ask whether one can take advantage of this factorization in the
computation of the residue of $G=PQ$ with respect to the polynomial system
$F_{0},\ldots,F_{n}$. In the case $X_{\Sigma}=\mathbb{P}^{n}$, Jouanolou
proved that this is indeed possible by exploiting the duality between the
degrees $\delta-\nu$ and $\nu$ of $P$ and $Q$, respectively; see [Jou97,
Proposition 3.10.27]. In what follows, we generalize Jouanolou’s formula to a
general smooth projective toric variety $X_{\Sigma}$. For that purpose, we
will rely on toric Sylvester forms and the elimination matrices
$\mathbb{H}_{\delta-\nu}$ we introduced in Section 5.1. The new formulas we
obtain can be seen as an extension of the rational formula “à la Macaulay”
proved in [DK05, Corollary 3.4].

######  Remark 7.1.

We notice that in [Jou97], Jouanolou defines the residue as a map onto $A$,
and not in $K(A)$, by multiplying with $\operatorname{Res}_{\mathcal{A}}$ in
the image; see also [CDS97, Theorem 1.4] for a proof that the product of the
residue and the resultant lies in $A$.

Let $\mathbb{H}_{\delta-\nu}$ be an elimination matrix that satisfies the
assumptions of Theorem 5.2 ii), and let $\mathcal{H}_{\delta-\nu}$ be a
maximal minor of $\mathbb{H}_{\delta-\nu}$ which contains the entire block
built with Sylvester forms. Now, being given two generic forms $P\in C_{\nu}$
and $Q\in C_{\delta-\nu}$, we consider the matrix

$\Theta_{\delta-\nu}=\begin{pmatrix}\mathcal{H}_{\delta-\nu}&\mathbf{q}\\\
\begin{matrix}0&\mathbf{p}^{T}\end{matrix}&0\end{pmatrix}$ (7.1)

where $\mathbf{p}$, respectively $\mathbf{q}$, stands for the vector of
coefficients of $P$, respectively $Q$. Recall that by construction of the
matrix $\mathbb{H}_{\delta-\nu}$, the matrix $\mathcal{H}_{\delta-\nu}$ is
built as the join of a Macaulay-type block-matrix and another column-block
matrix built from Sylvester forms. Thus, the row $\mathbf{p}^{T}$ is aligned
with the second column-block, built from Sylvester forms, of
$\mathcal{H}_{\delta-\nu}$; see Example 5.4 for an illustration.

As a first result, one proves that the residue of the product of two monomials
can be computed as a quotient of determinants. In what follows, We denote by
$\mathcal{H}_{\alpha,\beta}$ the submatrix of $\mathcal{H}_{\delta-\nu}$ that
is obtained by deleting the column corresponding to the monomial
$x^{\alpha}\in C_{\nu}$ and the row corresponding to the monomial
$x^{\beta}\in C_{\delta-\nu}$.

######  Lemma 7.1.

Let $F_{0},\dots,F_{n}$ be a system of homogeneous polynomials in $C$ as in
(2.4), then for any pair of monomials $x^{\alpha}\in C_{\nu}$ and
$x^{\beta}\in C_{\delta-\nu}$,

$\operatorname{Residue}_{F}(x^{\alpha+\beta})=(-1)^{\alpha+\beta}\frac{\det(\mathcal{H}_{\alpha,\beta})}{\det(\mathcal{H}_{\delta-\nu})}.$

###### Proof.

Let $H^{\beta}$ be the matrix obtained by multiplying the row of
$\det(\mathcal{H}_{\delta-\nu})$ corresponding to $x^{\beta}$ by the monomial
$x^{\beta}$ itself. Then, by expanding the determinant along this row, one
gets:

$x^{\alpha}x^{\beta}\det(\mathcal{H}_{\delta-\nu})=x^{\alpha}\det(H^{\beta})=x^{\alpha}(\sum
G_{i}F_{i}+\sum_{\alpha^{\prime}\in
C_{\nu}}c_{\alpha^{\prime},\beta}\operatorname{Sylv}_{\alpha^{\prime}})=\sum
x^{\alpha}G_{i}F_{i}+c_{\alpha,\beta}\operatorname{Sylv}_{0}.$

Taking residues at both sides, we deduce that

$\operatorname{Residue}_{F}(x^{\alpha+\beta})\det(\mathcal{H}_{\delta-\nu})=(-1)^{\alpha+\beta}c_{\alpha,\beta}.$

Finally, from the expansion of the determinant $\det(H^{\beta})$, one sees
immediately that $c_{\alpha,\beta}=\det(\mathcal{H}_{\alpha,\beta})$. ∎

We are now ready to prove the claimed formula for the residue of the product
of two forms.

######  Theorem 7.1.

Let $F_{0},\dots,F_{n}$ be a system of homogeneous polynomials in $C$ as in
(2.4), then for any pair of forms $P\in C_{\nu}$ and $Q\in C_{\delta-\nu}$,

$\operatorname{Residue}_{F}(PQ)=\frac{\det(\Theta_{\delta-\nu})}{\det(\mathcal{H}_{\delta-\nu})}.$

###### Proof.

Write $P=\sum_{x^{\alpha}\in C_{\nu}}p_{\alpha}x^{\alpha}$ and
$Q=\sum_{x^{\beta}\in C_{\delta-\nu}}q_{\beta}x^{\beta}$. Then, by linearlity
of residues, we have:

$\operatorname{Residue}_{F}(PQ)=\sum_{x^{\alpha}\in C_{\nu}\>x^{\beta}\in
C_{\delta-\nu}}p_{\alpha}q_{\beta}\operatorname{Residue}_{F}(x^{\alpha+\beta})=\frac{\sum_{\alpha,\beta}(-1)^{\alpha+\beta}p_{\alpha}q_{\beta}\det(\mathcal{H}_{\alpha,\beta})}{\det(\mathcal{H}_{\delta-\nu})}.$

The numerator is precisely the expansion of the determinant
$\det(\Theta_{\delta-\nu})$ of the matrix defined in (7.1), with respect to
the last row and column. ∎

######  Example 7.1.

In Example 5.1, the elimination matrix $\mathbb{H}_{\alpha}$ for
$\alpha=(2,1)$ is square, therefore we take

$\mathcal{H}_{(2,1)}=\mathbb{H}_{(2,1)}=\begin{pmatrix}a_{0}&b_{0}&c_{0}&\left[013\right]&\left[023\right]\\\
a_{1}&b_{1}&c_{1}&\left[023\right]+[014]&[024]+[123]\\\
a_{2}&b_{2}&c_{2}&[024]&[124]\\\ a_{3}&b_{3}&c_{3}&0&0\\\
a_{4}&b_{4}&c_{4}&0&0\end{pmatrix}$

Let $P=p_{0}z_{1}+p_{1}x_{1}$ and
$Q=q_{0}z_{1}^{2}z_{2}+q_{1}z_{1}z_{2}x_{1}+q_{2}z_{2}x_{1}^{2}+q_{3}z_{1}x_{2}+q_{4}x_{1}x_{2}$
be homogeneous forms in $C_{(1,0)}$ and $C_{(2,1)}$, respectively, then

$\Theta_{(2,1)}=\begin{pmatrix}a_{0}&b_{0}&c_{0}&\left[013\right]&\left[023\right]&q_{0}\\\
a_{1}&b_{1}&c_{1}&\left[023\right]+[014]&[024]+[123]&q_{1}\\\
a_{2}&b_{2}&c_{2}&[024]&[124]&q_{2}\\\ a_{3}&b_{3}&c_{3}&0&0&q_{3}\\\
a_{4}&b_{4}&c_{4}&0&0&q_{4}\\\ 0&0&0&p_{0}&p_{1}&0\end{pmatrix}.$

and aplying Theorem 7.1 we deduce that
$\operatorname{Residue}_{F}(PQ)=\frac{\det(\Theta_{(2,1)})}{\det(\mathcal{H}_{(2,1)})}$.
For the sake of comparison, let us examine the formula we obtain by developing
the product of $P$ and $Q$. In this case, we apply Theorem 7.1 with
$\delta=(3,1)$ and $\nu=0$, so we have to consider the matrix $\Theta_{(3,1)}$
which is of the form:

$\Theta_{(3,1)}=\begin{pmatrix}a_{0}&0&b_{0}&0&c_{0}&0&[130]&p_{0}q_{0}\\\
a_{1}&a_{0}&b_{1}&b_{0}&c_{1}&c_{0}&[230]&p_{0}q_{1}+p_{1}q_{0}\\\
a_{2}&a_{1}&b_{2}&b_{1}&c_{2}&c_{1}&0&p_{0}q_{2}+p_{1}q_{1}\\\
0&a_{2}&0&b_{2}&0&c_{2}&0&p_{1}q_{2}\\\
a_{3}&0&b_{3}&0&c_{3}&0&[430]&p_{0}q_{3}\\\
a_{4}&a_{3}&b_{4}&b_{3}&c_{4}&c_{3}&0&p_{0}q_{4}+p_{1}q_{3}\\\
0&a_{4}&0&b_{4}&0&c_{4}&0&p_{1}q_{4}\\\ 0&0&0&0&0&0&1&0\end{pmatrix}$

since

$PQ=p_{0}q_{0}z_{1}^{3}z_{2}+(p_{0}q_{1}+p_{1}q_{0})z_{1}^{2}z_{2}x_{1}+(p_{0}q_{2}+p_{1}q_{1})z_{1}z_{2}x_{1}^{2}+p_{0}q_{3}z_{1}^{2}x_{2}\\\
+(p_{0}q_{4}+p_{1}q_{3})z_{1}x_{1}x_{2}+p_{1}q_{2}z_{2}x_{1}^{3}+p_{1}q_{4}x_{1}^{2}x_{2}.$

The expansion of the determinant of $\Theta_{(3,1)}$ with respect to the last
row leads to the same formula as in [DK05, Corollary 3.4].

## Acknowledgments

The authors received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No
860843. They are grateful to Matías R. Bender, Marc Chardin and Carlos
D’Andrea for interesting discussions. They also thank Ioannis Z. Emiris,
Christos Konaxis and Elias Tsigaridas for their support as advisors of the
second author.

## References

  * [Alt+20] Klaus Altmann, Jarosław Buczyński, Lars Kastner and Anna-Lena Winz  “Immaculate line bundles on toric varieties”  In _Pure and Applied Mathematics_ 16.4, 2020, pp. 1141–1217  DOI: 10.4310/PAMQ.2020.v16.n4.a12
  * [BB11] Victor V. Batyrev and Lev A. Borisov  “On Calabi-Yau complete intersections in toric varieties”  In _Higher Dimensional Complex Varieties_ De Gruyter, 2011, pp. 39–66  DOI: doi:10.1515/9783110814736.39
  * [BC93] Victor V. Batyrev and David A. Cox  “On the Hodge structure of projective hypersurfaces in toric varieties”  In _Duke Mathematical Journal_ 75, 1993, pp. 293–338
  * [BCN22] Laurent Busé, Marc Chardin and Navid Nemati  “Multigraded Sylvester forms, duality and elimination matrices”  In _Journal of Algebra_ 609, 2022, pp. 514–546  DOI: https://doi.org/10.1016/j.jalgebra.2022.06.022
  * [BCP23] Laurent Busé, Fabrizio Catanese and Elisa Postinghel  “Algebraic curves and surfaces: a history of shapes” To appear  Springer, 2023
  * [Bec02] Matthias Beck  “Multidimensional Ehrhart Reciprocity”  In _Journal of Combinatorial Theory, Series A_ 97.1, 2002, pp. 187–194  DOI: https://doi.org/10.1006/jcta.2001.3220
  * [Ben+21] Matías R. Bender, Jean-Charles Faugère, Angelos Mantzaflaris and Elias Tsigaridas  “Koszul-type determinantal formulas for families of mixed multilinear systems”  In _SIAM Journal on Applied Algebra and Geometry_ 5.4, 2021, pp. 589–619  DOI: 10.1137/20M1332190
  * [BJ14] Laurent Busé and Jean-Pierre Jouanolou  “On the discriminant scheme of homogeneous polynomials”  In _Mathematics in Computer Science_ 8.2, Special Issue in Computational Algebraic Geometry  Springer, 2014, pp. 175–234  DOI: 10.1007/s11786-014-0188-7
  * [BM02] Victor V. Batyrev and Evgeny Materov  “Toric Residues and Mirror Symmetry”  In _Mosc. Math. J._ 2, 2002  DOI: 10.17323/1609-4514-2002-2-3-435-475
  * [Bot11] Nicolás Botbol  “The implicit equation of a multigraded hypersurface”  In _Journal of Algebra_ 348.1, 2011, pp. 381–401  DOI: https://doi.org/10.1016/j.jalgebra.2011.09.019
  * [BT22] Matías R. Bender and Simon Telen  “Toric eigenvalue methods for solving sparse polynomial systems”  In _Mathematics of Computation_ 91, 2022, pp. 2397–2429  DOI: 10.1090/mcom/3744
  * [CCD97] Eduardo Cattani, David A. Cox and Alicia Dickenstein  “Residues in toric varieties”  In _Compositio Mathematica_ 108.1 London Mathematical Society, 1997, pp. 35–76  DOI: 10.1023/A:1000180417349
  * [CDS97] Eduardo Cattani, Alicia Dickenstein and Bernd Sturmfels  “Residues and resultants”  In _Journal of Mathematical Sciences_ 5, 1997, pp. 119–148
  * [CE22] Carles Checa and Ioannis Emiris  “A greedy approach to the Canny-Emiris formula”  In _Proceedings of ISSAC ’22_ Villeneuve-d’Ascq, France: Association for Computing Machinery, 2022, pp. 283–291  DOI: 10.1145/3476446.3536180
  * [CE93] John F. Canny and Ioannis Z. Emiris  “An efficient algorithm for the sparse mixed resultant”  In _Applied algebra, algebraic algorithms and error-correcting codes_ 673, 1993, pp. 89–104  DOI: 10.1007/3-540-56686-4˙36
  * [Cha13] Marc Chardin  “Powers of ideals and the cohomology of stalks and fibers of morphisms”  In _Algebra and Number Theory_ 7.1, 2013, pp. 1–18  DOI: 10.2140/ant.2013.7-1
  * [CLS12] David A. Cox, John B. Little and Hal Schenck  “Toric Varieties”  In _Jahresbericht der Deutschen Mathematiker-Vereinigung_ 114.3, 2012, pp. 367–380  DOI: https://doi.org/10.1365/s13291-012-0048-9
  * [Cox95] David A. Cox  “The homogeneous coordinate ring of a toric variety”  In _Journal of Algebraic Geometry_ , 1995, pp. 17–50
  * [Cox96] David A. Cox  “Toric residues”  In _Arkiv for Matematik_ 34, 1996, pp. 73–96  DOI: 10.1007/BF02559508
  * [CP93] John F. Canny and Paul Pedersen  “An algorithm for the Newton resultant”  USA: Cornell University, 1993
  * [DD00] Carlos D’Andrea and Alicia Dickenstein  “Explicit formulas for the multivariate resultant”  In _Journal of Pure and Applied Algebra_ 164, 2000, pp. 59–86
  * [DE01] Carlos D’Andrea and Ioannis Z. Emiris  “Hybrid sparse resultant matrices for bivariate systems”, ISSAC ’01  London, Ontario, Canada: Association for Computing Machinery, 2001, pp. 24–31  URL: https://doi.org/10.1145/384101.384106
  * [Dem70] Michel Demazure  “Sous-groupes algébriques de rang maximum du groupe de Cremona”  In _Annales scientifiques de l’École Normale Supérieure_ 4e série, 3.4 Elsevier, 1970, pp. 507–588  DOI: 10.24033/asens.1201
  * [DJS22] Carlos D’Andrea, Gabriela Jeronimo and Martín Sombra  “The Canny-Emiris conjecture for the sparse resultant”  In _Foundations of Computational Mathematics_ , 2022  DOI: 10.1007/s10208-021-09547-3
  * [DK05] Carlos D’Andrea and Amit Khetan  “Macaulay style formulas for toric residues”  In _Compositio Mathematica_ 141.3 London Mathematical Society, 2005, pp. 713–728  DOI: 10.1112/S0010437X05001326
  * [EM99] Ioannis Z. Emiris and Bernard Mourrain  “Matrices in elimination theory”  In _Journal of Symbolic Computation_ 28.1, 1999, pp. 3–44  DOI: https://doi.org/10.1006/jsco.1998.0266
  * [EMS00] David Eisenbud, Mircea Mustata and Michael Stillman  “Cohomology on toric varieties and local cohomology with monomial support”  In _J. Symb. Comput._ 29, 2000, pp. 583–600  DOI: 10.1006/jsco.1999.0326
  * [GKZ94] Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky  “Discriminants, resultants, and multidimensional determinants”  Birkhäuser Boston, MA, 1994  DOI: https://doi.org/10.1007/978-0-8176-4771-1
  * [Gro57] Alexander Grothendieck  “Sur quelques points d’algèbre homologique, I”  In _Tohoku Mathematical Journal_ 9.2 Tohoku University, Mathematical Institute, 1957, pp. 119 –221  DOI: 10.2748/tmj/1178244839
  * [Har77] Robin Hartshorne  “Algebraic geometry” Graduate Texts in Mathematics, No. 52  New York: Springer-Verlag, 1977, pp. xvi+496
  * [Jou91] Jean-Pierre Jouanolou  “Le formalisme du résultant”  In _Adv. Math._ 90.2, 1991, pp. 117–263
  * [Jou95] Jean-Pierre Jouanolou  “Aspects invariants de l’elimination”  In _Adv. Math._ 114.1, 1995
  * [Jou97] Jean-Pierre Jouanolou  “Formes d’inertie et résultant: Un formulaire”  In _Advances in Mathematics_ 126, 1997, pp. 119–250  DOI: 10.1006/aima.1996.1609
  * [KS05] Amit Khetan and Ivan Soprounov  “Combinatorial construction of toric residues.”  In _Annales de l’institut Fourier_ 55.2 Association des Annales de l’Institut Fourier, 2005, pp. 511–548  URL: http://eudml.org/doc/116199
  * [MS03] Diane Maclagan and Gregory G. Smith  “Uniform bounds on multigraded regularity”  In _Journal of Algebraic Geometry_ 14, 2003, pp. 137–164
  * [Sop07] Ivan Soprunov  “Global residues for sparse polynomial systems”  In _Journal of Pure and Applied Algebra_ 209.2 Elsevier BV, 2007, pp. 383–392  DOI: 10.1016/j.jpaa.2006.06.012
  * [Tel20] Simon Telen  “Solving systems of polynomial equations”, 2020
  * [WZ92] Jerzy Weyman and Andrei Zelevinsky  “Determinantal formulas for multigraded resultants.”  In _Djursholm: Institut Mittag-Leffler_ 36, 1992  DOI: 10.1016/S0747-7171(03)00086-5
  * [Zar37] Oscar Zariski  “Generalized weight properties of the resultant of $n+1$ polynomials in $n$ indeterminates”  In _Trans. Amer. Math. Soc._ 41.2, 1937, pp. 249–265  DOI: 10.2307/1989621