# Koszul-type determinantal formulas for families of mixed multilinear systems

Matías R. Bender  Technische Universität Berlin, Institut für Mathematik,
Berlin, Germany (mbender@math.tu-berlin.de). Jean-Charles Faugère CryptoNext
Security and Sorbonne Université, CNRS, INRIA, Laboratoire d’Informatique de
Paris 6, LIP6, Équipe PolSys, 4 place Jussieu, F-75005, Paris, France (jean-
charles.faugere@inria.fr). Angelos Mantzaflaris Inria Sophia-Antipolis
Méditerranée, Université Côte d’Azur, 06902, France
<EMAIL_ADDRESS>Elias Tsigaridas Inria Paris, Institut de
Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Université and Paris
Université, 4 place Jussieu, F-75005, Paris, France
<EMAIL_ADDRESS>
###### Abstract

Effective computation of resultants is a central problem in elimination theory
and polynomial system solving. Commonly, we compute the resultant as a
quotient of determinants of matrices and we say that there exists a
determinantal formula when we can express it as a determinant of a matrix
whose elements are the coefficients of the input polynomials. We study the
resultant in the context of mixed multilinear polynomial systems, that is
multilinear systems with polynomials having different supports, on which
determinantal formulas were not known. We construct determinantal formulas for
two kind of multilinear systems related to the Multiparameter Eigenvalue
Problem (MEP): first, when the polynomials agree in all but one block of
variables; second, when the polynomials are bilinear with different supports,
related to a bipartite graph. We use the Weyman complex to construct Koszul-
type determinantal formulas that generalize Sylvester-type formulas. We can
use the matrices associated to these formulas to solve square systems without
computing the resultant. The combination of the resultant matrices with the
eigenvalue and eigenvector criterion for polynomial systems leads to a new
approach for solving MEP.

Key words. Resultant matrix, Multilinear polynomial, Weyman complex,
Determinantal formula, Koszul-type matrix, Multiparameter eigenvalue problem

AMS subject classifications. 13P15 14Q20 15A18

##  1 Introduction

One of the main questions in (computational) algebraic geometry is to decide
efficiently when an overdetermined polynomial system has a solution over a
projective variety. The resultant answers this question. The resultant is a
multihomogeneous polynomial in the coefficients of the polynomials of the
system that vanishes if and only if the system has a solution. We can also use
it to solve square systems. When we restrict the supports of the input
polynomials to make them sparse, we have an analogous concept called the
sparse resultant [28]. The sparse resultant is one of the few tools we can use
to solve systems taking into account the sparsity of the support of the
polynomials. Hence, its efficient computation is fundamental in computational
algebraic geometry.

We are interested in the computation of the multiprojective resultant, as it
is defined in [17, 20, 43], of sparse systems given by a particular kind of
multilinear polynomials. To define the multiprojective resultant as a single
polynomial, we restrict ourselves to systems where the number of equations is
one greater than the dimension of the ambient multiprojective space. In what
follows, we refer to this specific situation as an overdetermined system111For
general overdetermined systems, there exists the concept of _resultant system_
, see [51, Sec. 16,5]. . In general, we compute the resultant of a polynomial
system $(f_{0},\dots,f_{n})$ as a quotient of determinants of two matrices
whose elements are polynomials in the coefficients of the input polynomials
[11, 12, 15, 16, 33, 36, 37]; thus the best we can hope for are linear
polynomials. A classical example of such a matrix is the Macaulay matrix,
which represents a map $(g_{0},\dots,g_{n})\mapsto\sum_{i}g_{i}\,f_{i}$, where
each $g_{i}$ is a polynomial in a finite dimensional vector space. In this
case, we say that we have a _Sylvester-type_ formula. Other classical formulas
include _Bézout-_ and _Dixon-type_ ; nevertheless, the elements of the
corresponding matrices are not linear anymore. We refer to [25] and references
therein for details.

When we can compute the resultant as the determinant of a matrix we say that
we have a determinantal formula. Besides general constructions that express
any multivariate polynomial as a determinant of a matrix, see for example [35,
49], we are interested in formulas such that the row/column dimension of the
corresponding matrix depends linearly on the degree of the resultant. The
existence of such formulas is not known in general. When we consider unmixed
multihomogeneous systems, that is when every polynomial has the same support,
these formulas are well studied, e.g., [11, 18, 47, 54]. However, when the
supports are not the same, that is in the case of mixed multihomogeneous
systems, there are very few results. We know determinantal formulas for scaled
multihomogeneous systems [22], in which case the supports are scaled copies of
one of them, for bivariate tensor-product polynomial systems [9], and for
bilinear systems with two different supports [5]. One tool to obtain such
formulas is using the Weyman complex [52]. For an introduction to this complex
we refer to [53, Sec. 9.2] and [28, Sec. 2.5.C, Sec. 3.4.E].

Resultant computations are also useful in solving $0$-dimensional square
polynomial systems, say $(f_{1},\dots,f_{N})$, taking into account the
sparsity; here ”square” refers to systems having $N$ polynomials in $N$
variables. For example we can use the $u$-resultant or hide a variable; we
refer to [14, Ch. 3] for a general introduction. Whenever a _Sylvester-type
formula_ is available, through the resultant matrix, we obtain a matrix
representing the multiplication map by a polynomial $f_{0}$ in
$\mathbb{K}[\boldsymbol{x}]/\langle f_{1},\dots,f_{N}\rangle$. Then, we solve
the system $(f_{1},\dots,f_{N})$ by computing the eigenvalues and eigenvectors
of this matrix, e.g., [4, 21]. The eigenvalues correspond to the evaluations
of $f_{0}$ at the solutions of the system. From the eigenvectors, at least
when there are no multiplicities, we can recover the coordinates of the
solutions. For a generalization of this approach to a broader class of
resultant matrices, that encapsulates Sylvester-type matrices as a special
case, we refer to [5].

###  1.1 Multilinear polynomial systems

We focus on computing determinantal formulas for mixed multilinear polynomial
systems. Besides their mathematical importance, as they are the first non-
trivial case of polynomial systems beyond the linear ones, multilinear systems
are also ubiquitous in applications, e.g., cryptography [26, 34] and game
theory [39].

For $A,B\in{\mathbb{N}}$ let $\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{A}$,
$\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{B}$ be blocks of variables. We
present various Koszul-type determinantal formulas (related to the maps in the
Koszul complex, see Def. 2.14) for the following two kinds of mixed
multilinear polynomials systems $(f_{0},f_{1}\dots f_{N})$:

  * •

star multilinear systems: these are polynomial systems $(f_{1},\dots,f_{N})$,
where for each $f_{k}$, there is a $j_{k}\in[B]$ such that

$f_{k}\in\mathbb{K}[\boldsymbol{X}_{1}]_{1}\otimes\dots\otimes\mathbb{K}[\boldsymbol{X}_{A}]_{1}\otimes\mathbb{K}[\boldsymbol{Y}_{j_{k}}]_{1},$

  * •

bipartite bilinear systems: these are polynomial systems
$(f_{1},\dots,f_{N})$, where for each $f_{k}$, there are $i_{k}\in[A]$ and
$j_{k}\in[B]$ such that

$f_{k}\in\mathbb{K}[\boldsymbol{X}_{i_{k}}]_{1}\otimes\mathbb{K}[\boldsymbol{Y}_{j_{k}}]_{1}.$

To make the system overdetermined and so, to consider its resultant, we
complement it with several types of multilinear polynomials $f_{0}$ (see the
beginning of sections 3 and 4).

Our first main contribution is the theorem below which is an extract of Thm.
3.7 and 4.4.

######  Theorem 1.1.

Let $\boldsymbol{f}:=(f_{1},\dots,f_{N})$ be a _star multilinear system_ (Def.
3.1) or a _bipartite bilinear system_ (Def. 4.1). Then, for certain choices of
multilinear polynomials $f_{0}$ (we present them at the beginning of Sec. 3
and 4), there is a square matrix $M$ such that,

  * •

The resultant of $\boldsymbol{f}$ agrees with the determinant of the matrix,
$\mathtt{res}(\boldsymbol{f})=\pm\det(M)$.

  * •

The number of columns/rows of $M$ is $degree(\mathtt{res}(\boldsymbol{f}))$
and its elements are coefficients of $\boldsymbol{f}$, possibly with a sign
change.

The matrix $M$ corresponds to a _Koszul-type determinantal formula_ (Def.
2.14) for $\boldsymbol{f}$.

The size of the resultant matrix and the degree of the resultant depend on the
multidegree of $f_{0}$. We relate the (expected) number of solutions of
$(f_{1},\dots,f_{N})$ to the degree of the resultant of
$(f_{0},f_{1},\dots,f_{N})$. For star multilinear systems, we present closed
formulas for the expected number of solutions of the system
$(f_{1},\dots,f_{N})$ and the size of the matrices; we also express the size
of the matrix in terms of the number of solutions. Our techniques to obtain
determinantal formulas exploit the properties and the parametrization, through
a carefully chosen degree vector, of the Weyman complex and are of independent
interest. These results generalize the ones in [5] that correspond to the case
$(A=1,B=2)$.

###  1.2 Multiparameter Eigenvalue Problem

A motivating application for the systems and the determinantal formulas that
we study comes from the _multiparameter eigenvalue problem_ (MEP). We can
model MEP using _star multilinear systems_ or _bipartite bilinear systems_.
The resultant matrices that we construct together with the eigenvalue and
eigenvector criterion for polynomial systems, e.g., [13], lead to a new
approach for solving MEP.

MEP generalizes the classical eigenvalue problem. It arises in mathematical
physics as a way of solving ordinary and partial differential equations when
we can use separation of variables (Fourier method) to solve boundary
eigenvalue problems. Its applications, among others, include the Spectral and
the Sturm-Liouville theory [2, 3, 29, 32, 50]. MEP allows us to solve
different eigenvalue problems, e.g., the polynomial and the quadratic two-
parameter eigenvalue problems [30, 40]. It is an old problem; its origins date
from the 1920’ in the works of R. D. Carmichael [10] and A. J. Pell [42].

The precise definition of the problem is as follows. Assume
$\alpha\in\mathbb{N}$, $\beta_{1},\dots,\beta_{\alpha}\in\mathbb{N}$, and
consider matrices $\\{M^{(i,j)}\\}_{0\leq
j\leq\alpha}\in\mathbb{K}^{(\beta_{i}+1)\times(\beta_{i}+1)}$, where $0\leq
i\leq\alpha$. The MEP consists in finding
$\boldsymbol{\lambda}=(\lambda_{0},\dots,\lambda_{\alpha})\in\mathbb{P}^{\alpha}(\mathbb{K})$
and
$\boldsymbol{v_{1}}\in~{}\mathbb{P}^{\beta_{1}}(\mathbb{K}),\dots,\boldsymbol{v_{\alpha}}\in\mathbb{P}^{\beta_{\alpha}}(\mathbb{K})$
such that

$\Big{(}\sum\nolimits_{j=0}^{\alpha}\lambda_{j}\,M^{(1,j)}\Big{)}\,\boldsymbol{v_{1}}=0\
,\,\dots,\
\Big{(}\sum\nolimits_{j=0}^{\alpha}\lambda_{j}\,M^{(\alpha,j)}\Big{)}\,\boldsymbol{v_{\alpha}}=0,$
(1)

where $\mathbb{K}$ is an algebraically closed field and
$\mathbb{P}^{n}(\mathbb{K})$ is the (corresponding) projective space of
dimension $n\in\mathbb{N}$. We refer to $\boldsymbol{\lambda}$ as an
eigenvalue, $(\boldsymbol{v_{1}},\dots,\boldsymbol{v_{\alpha}})$ as an
eigenvector, and to
$(\boldsymbol{\lambda},\boldsymbol{v_{1}},\dots,\boldsymbol{v_{\alpha}})$ as
an eigenpair. For $\alpha=1$, MEP is the _generalized eigenvalue problem_.

To exploit our tools we need to write MEP as a mixed square bilinear system.
For this we introduce the variables
$\boldsymbol{X}_{1}=(x_{0},\dots,x_{\alpha})$ to represent the multiparameter
eigenvalues and, for each $1\leq i\leq\alpha$, the vectors
$\boldsymbol{Y}_{t}=(y_{i,0},\dots,y_{i,\beta_{i}})$ to represent the
eigenvectors. This way, we obtain a bilinear system
$\boldsymbol{F}=(f_{1,0},\dots,f_{\alpha,\beta_{\alpha}})$, where for each
$1\leq t\leq\alpha$,

$\displaystyle\small\begin{array}[]{c
}\displaystyle\left(\sum_{j=0}^{\alpha}x_{j}\,M^{(t,j)}\right)\cdot\begin{pmatrix}y_{t,0}\\\
y_{t,1}\\\ \vdots\\\
y_{t,\beta_{t}}\end{pmatrix}=\begin{pmatrix}\displaystyle\sum_{i=0}^{\beta_{t}}\sum_{j=0}^{\alpha}M^{(t,j)}_{i,0}\,x_{j}\,y_{t,i}\\\
\vdots\\\
\displaystyle\sum_{i=0}^{\beta_{t}}\sum_{j=0}^{\alpha}M^{(t,j)}_{i,\beta_{t}}\,x_{j}\,y_{t,i}\\\
\end{pmatrix}=\begin{pmatrix}\\\\[0.0pt] f_{t,0}\\\\[10.0pt]
\vdots\\\\[10.0pt] f_{t,\beta_{t}}\\\\[10.0pt] \end{pmatrix}\end{array}$ (3)

and, for each $1\leq t\leq\alpha$,
$f_{t,0},\dots,f_{t,\beta_{t}}\in\mathbb{K}[\boldsymbol{X}_{1}]_{1}\otimes\mathbb{K}[\boldsymbol{Y}_{t}]_{1}$.
In this formulation, the system in (3) is a particular case of a star
multilinear system (Def. 3.1) with $A=1$ and $B=\alpha$, or a particular case
of bipartite bilinear system (Def. 4.1) with $A=1$ and $B=\alpha$. There is a
one to one correspondence between the eigenpairs of MEP and solutions of
$\boldsymbol{F}$, that is

$\begin{array}[]{c}(\boldsymbol{\lambda},\boldsymbol{v}_{1},\dots,\boldsymbol{v}_{\alpha})\text{
is an}\\\ \text{eigenpair of
}\\{M^{(i,j)}\\}.\end{array}\iff\\!\\!\\!\\!\begin{array}[]{c}(\boldsymbol{\lambda},\boldsymbol{v}_{1},\dots,\boldsymbol{v}_{\alpha})\in\mathbb{P}^{\alpha}(\mathbb{K})\times\mathbb{P}^{\beta_{1}}(\mathbb{K})\times\dots\times\mathbb{P}^{\beta_{\alpha}}(\mathbb{K})\\\
\text{ and
}\boldsymbol{F}(\boldsymbol{\lambda},\boldsymbol{v}_{1},\dots,\boldsymbol{v}_{\alpha})=0.\end{array}$

The standard method to solve MEP is Atkinson’s _Delta method_ [3, Ch. 6, 8].
For each $0\leq k\leq\alpha$, it considers the overdetermined system
$\boldsymbol{F}_{k}$ resulting from $\boldsymbol{F}$ (3) by setting $x_{k}=0$.
Then, it constructs a matrix $\Delta_{k}$ which is nonsingular if and only if
$\boldsymbol{F}_{k}$ has no solutions [3, Eq. 6.4.4]. Subsequently, it applies
linear algebra operations to these matrices to solve the MEP $\boldsymbol{F}$
[3, Thm. 6.8.1] . It turns out that the matrices $\Delta_{k}$ are
determinantal formulas for the resultants of the corresponding overdetermined
systems $\boldsymbol{F}_{k}$. The elements of the matrices of the
determinantal formulas $\Delta_{k}$ are polynomials of degree $\alpha$ in the
elements of the matrices $M^{(i,j)}$ [3, Thm. 8.2.1]. The Delta method can
only solve _nonsingular MEPs_ ; these are MEPs where there exists a finite
number of eigenvalues [3, Ch. 8]. The main computational disadvantage of
Atkinson’s Delta method is the cost of computing the matrices $\Delta_{k}$. To
compute these matrices one needs to consider multiple Kronecker products
corresponding to the Laplace expansion of a determinant of size
$\alpha\times\alpha$. The interested reader can find more details in [3, Sec.
6.2].

Besides Atkinson’s Delta method, there are recent algorithms such as the
_diagonal coefficient homotopy method_ [19] and the _fiber product homotopy
method_ [44] which exploit homotopy continuation methods. These methods seems
to be slower than the Delta method, but they can tackle MEPs of bigger size as
they avoid the construction of the matrices $\Delta_{k}$. The Delta method and
the homotopy approaches can only compute the _regular_ eigenpairs of the MEP,
that is, those where the eigenpair is an isolated solution of (3). In contrast
to the Delta method, experimentally and in some cases, the _fiber product
homotopy method_ can also solve _singular MEPs_ , see [44, Sec. 10]. We can
also use general purpose polynomial system solving algorithms, that exploit
sparsity, to tackle MEP. We refer reader to [27], see also [46], for an
algorithm to solve unmixed multilinear systems using Gröbner bases, and to
[23, 24] using resultants. We also refer to [6, 7] for an algorithm based on
Gröbner bases to solve square mixed multihomogeneous systems and to [8, 48]
for a numerical algorithm to solve these systems using eigenvalue
computations. However, these generic approaches do not fully exploit the
structure of the problem.

Our second main contribution is a new approach to solve MEP based on the
determinantal formulas that we develop (Alg. 1). We present a novel way to
compute the $\Delta$ matrices by avoiding the expansion to minors. Moreover,
the resultant matrix from which this construction emanates has elements that
are linear in the elements of $M^{(i,j)}$, is highly structured, and with few
non-zero elements (Rem. 5.2). As the Delta method and the homotopy algorithms,
it can only solve _nonsingular MEPs_ and recover the _regular eigenvalues_.
Contrary to the general purpose approaches, it fully exploits the structure of
the system and its complexity relates to the number of eigenpairs. Our
approach involves the computation of (standard) eigenvalues and eigenvectors
to recover the solutions of the MEP, as the classical symbolic-numeric
methods, e.g., [13]. In particular, our method works with exact coefficients
as well as with approximations. The code for solving MEP using these resultant
matrices is freely available at
https://mbender.github.io/multLinDetForm/sylvesterMEP.m.

#### Organization of the paper

In Sec. 2.1 we define the multihomogeneous systems and introduce some
notation. Later, in Sec. 2.2, we introduce the multihomogeneous resultant. In
Sec. 2.3, we introduce Weyman complexes and then, in Sec. 2.4, we explain the
Koszul-type formulas. In Sec. 3, we define the _star multilinear systems_ and
we construct _Koszul-type determinantal formulas_ for multihomogeneous systems
involving them, and in Sec. 3.2 we study the number of solutions of the
systems and we compare them with the sizes of the determinant. We also present
an example in Sec. 3.3. In Sec. 4, we define the _bipartite bilinear systems_
and construct _Koszul-type determinantal formulas_ for multihomogeneous
systems involving them. Finally, in Sec. 5, we present an algorithm and an
example for solving MEP using our _determinantal formulas_.

##  2 Preliminaries

For a number $N\in\mathbb{N}$ we use the abbreviation $[N]=\\{1,\dots,N\\}$.

###  2.1 Multihomogeneous systems

Let $\mathbb{K}$ be an algebraically closed field, $\mathbb{K}^{m}$ a vector
space, and $(\mathbb{K}^{m})^{*}$ its dual space, where $m\in{\mathbb{N}}$.
Let $q\in\mathbb{N}$ and consider $q$ positive natural numbers,
$n_{1},\dots,n_{q}\in\mathbb{N}$. For each $i\in[q]$, we consider the
following sets of $n_{i}+1$ variables

$\boldsymbol{x_{i}}:=\\{x_{i,0},\dots,x_{i,n_{i}}\\}\qquad\text{and}\qquad\boldsymbol{\partial
x_{i}}:=\\{\partial x_{i,0},\dots,\partial x_{i,n_{i}}\\}.$

We identify the polynomial algebra $\mathbb{K}[\boldsymbol{x_{i}}]$ with the
symmetric algebra of the vector space $\mathbb{K}^{n_{i}+1}$ and the algebra
$\mathbb{K}[\boldsymbol{\partial x_{i}}]$ with the symmetric algebra of
$(\mathbb{K}^{n_{i}+1})^{*}$. That is

$\mathbb{K}[\boldsymbol{x_{i}}]\cong
S\left(\mathbb{K}^{n_{i}+1}\right)=\bigoplus_{d\in\mathbb{Z}}S_{i}(d)\qquad\text{and}\qquad\mathbb{K}[\boldsymbol{\partial
x_{i}}]\cong
S\left((\mathbb{K}^{n_{i}+1})^{*}\right)=\bigoplus_{d\in\mathbb{Z}}S^{*}_{i}(-d).$

Therefore, for each $i$, $S_{i}(d)$ corresponds to the $\mathbb{K}$-vector
space of polynomials in $\mathbb{K}[\boldsymbol{x_{i}}]$ of degree $d$ and
$S_{i}^{*}(-d)$ to the $\mathbb{K}$-vector space of polynomials in
$\mathbb{K}[\boldsymbol{\partial x_{i}}]$ of degree $d$. Note that if $d<0$,
then $S_{i}(d)=S_{i}^{*}(-d)=0$.

We identify the monomials in $\mathbb{K}[\boldsymbol{x_{i}}]$ and
$\mathbb{K}[\boldsymbol{\partial x_{i}}]$ with vectors in
$\mathbb{Z}^{n_{i}+1}$. For each
$\boldsymbol{\alpha}=(\alpha_{0},\dots,\alpha_{n_{i}})\in\mathbb{Z}^{n_{i}+1}$
we set
$\boldsymbol{x_{i}}^{\boldsymbol{\alpha}}:=\prod_{j=0}^{n_{i}}x_{i,j}^{\alpha_{j}},$
and $\boldsymbol{\partial
x_{i}}^{\boldsymbol{\alpha}}:=\prod_{j=0}^{n_{i}}\partial
x_{i,j}^{\alpha_{j}}.$ We consider the ${\mathbb{Z}}^{q}$-graded polynomial
algebra

$\mathbb{K}[\boldsymbol{\bar{x}}]:=\mathbb{K}[\boldsymbol{x_{1}}]\otimes\dots\otimes\mathbb{K}[\boldsymbol{x_{q}}]\cong\bigoplus_{(d_{1},\dots,d_{q})\in\mathbb{Z}^{q}}S_{1}(d_{1})\otimes\dots\otimes
S_{q}(d_{q}).$

Notice that for each $\boldsymbol{d}=(d_{1},\dots,d_{q})\in\mathbb{Z}^{q}$,
$\mathbb{K}[\boldsymbol{\bar{x}}]_{\boldsymbol{d}}$ is the $\mathbb{K}$-vector
space of the multihomogeneous polynomials of multidegree $\boldsymbol{d}$,
that is polynomials in $\mathbb{K}[\boldsymbol{\bar{x}}]$ having degree
$d_{i}$ with respect to the set of variables $\boldsymbol{x_{i}}$, for each
$i\in[q]$. We say that a polynomial $f\in\mathbb{K}[\boldsymbol{\bar{x}}]$ is
multihomogeneous of multidegree $\boldsymbol{d}\in\mathbb{Z}^{q}$, if
$f\in\mathbb{K}[\boldsymbol{\bar{x}}]_{\boldsymbol{d}}=S_{1}(d_{1})\otimes\dots\otimes
S_{q}(d_{q})$.

######  Example 2.1.

Consider the two blocks of variables
$\boldsymbol{x_{1}}\\!=\\{x_{1,0},x_{1,1}\\}$ and
$\boldsymbol{x_{2}}\\!=\\{x_{2,0},x_{2,1},x_{2,2}\\}$ and the polynomial
$x_{1,0}^{2}\,x_{1,1}\otimes x_{2,0}\,x_{2,1}+x_{1,0}\,x_{1,1}^{2}\otimes
x_{2,2}^{2}\in\mathbb{K}[\boldsymbol{x_{1}},\boldsymbol{x_{2}}]$. It is
multihomogeneous of multidegree $(3,2)$ and we write it simply as
$x_{1,0}^{2}\,x_{1,1}\,x_{2,0}\,x_{2,1}+x_{1,0}\,x_{1,1}^{2}\,x_{2,2}^{2}$.

Following standard notation, we write the monomials of
$\mathbb{K}[\boldsymbol{\bar{x}}]$ as
$\prod_{i=1}^{q}\boldsymbol{x_{i}}^{\boldsymbol{\alpha_{i}}}$ instead of
$\bigotimes_{i=1}^{q}\boldsymbol{x_{i}}^{\boldsymbol{\alpha_{i}}}$. We
identify these monomials with vectors in
$\mathbb{Z}^{n_{1}+1}\times\dots\times\mathbb{Z}^{n_{q}+1}$. For each
$\boldsymbol{\alpha}=(\boldsymbol{\alpha_{1}},\dots,\boldsymbol{\alpha_{q}})\in\mathbb{Z}^{n_{1}+1}\times\dots\times\mathbb{Z}^{n_{q}+1}$
we set
$\boldsymbol{x^{\alpha}}:=\prod_{i=1}^{q}\boldsymbol{x_{i}}^{\boldsymbol{\alpha}_{i}}.$
For each multidegree $\boldsymbol{d}\in\mathbb{Z}^{q}$, we denote by
$\mathcal{A}(\boldsymbol{d})$ the set of exponents of the monomials of
multidegree $\boldsymbol{d}$, that is
$\mathcal{A}(\boldsymbol{d})=\\{\boldsymbol{\alpha}\in\mathbb{Z}^{n_{1}+1}\times\dots\times\mathbb{Z}^{n_{q}+1}:\boldsymbol{x}^{\boldsymbol{\alpha}}\in\mathbb{K}[\boldsymbol{\bar{x}}]_{\boldsymbol{d}}\\}.$
The cardinality of $\mathcal{A}(\boldsymbol{d})$ is
$\\#\mathcal{A}(\boldsymbol{d})=\prod_{i=1}^{q}{d_{i}+n_{i}\choose n_{i}}.$

We fix $N=n_{1}+\dots+n_{q}$. Let
$\mathcal{P}:=\mathbb{P}^{n_{1}}\times\dots\times\mathbb{P}^{n_{q}}$ be a
multiprojective space over $\mathbb{K}$, where
$\mathbb{P}^{n_{i}}:=\mathbb{P}^{n_{i}}(\mathbb{K})$. A system of
multihomogeneous polynomials is a set of multihomogeneous polynomials in
$\mathbb{K}[\boldsymbol{\bar{x}}]$. We say that a system of multihomogeneous
polynomials $\\{f_{1},\dots,f_{r}\\}$ has a solution in $\mathcal{P}$, if
there is $\gamma\in\mathcal{P}$ such that $f_{i}(\gamma)=0$, for every $1\leq
i\leq r$. We call a system of multihomogeneous polynomials _square_ if $r=N$,
and _overdetermined_ if $r=N+1$. Generically, square multihomogeneous systems
have a finite number of solutions over $\mathcal{P}$, while the overdetermined
ones do not have solutions. The following proposition bounds the number of
solutions of square multihomogeneous polynomial systems.

######  Proposition 2.2 (Multihomogeneous Bézout bound, [45, Example 4.9]).

Consider a square multihomogeneous system
$\boldsymbol{f}:=\\{f_{1},\dots,f_{N}\\}$ of multidegrees
$\boldsymbol{d}_{1},\dots,\boldsymbol{d}_{N}\in\mathbb{N}^{q}$ with
$\boldsymbol{d}_{i}=(d_{i,1},\dots,d_{i,q})$, for each $1\leq i\leq N$. If
$\boldsymbol{f}$ has a finite number of solutions over $\mathcal{P}$, then
their number, counted with multiplicities, see [14, Sec. 4.2], is the
coefficient of the monomial $\prod_{i}Z_{i}^{n_{i}}$ in the polynomial
$\prod_{k=1}^{N}\left(\sum_{i=1}^{q}d_{k,i}\,Z_{i}\right).$ We refer to this
coefficient as the _multihomogeneous Bézout bound_ and we will write it as
$\mathtt{MHB}(\boldsymbol{d}_{1},\dots,\boldsymbol{d}_{N})$.

The multihomogeneous Bézout bound is generically tight, see [17, Thm. 1.11].

###  2.2 Multihomogeneous resultant

We fix $N+1$ multidegrees
$\boldsymbol{d_{0}},\boldsymbol{d_{1}},\dots,\boldsymbol{d_{N}}\in\mathbb{N}^{q}$.
To characterize the overdetermined multihomogeneous systems of such
multidegrees having solutions over $\mathcal{P}$, we parameterize them and we
introduce the resultant. The latter is a polynomial in the coefficients of the
polynomials of the system that vanishes if and only if the system has a
solution over $\mathcal{P}$. Our presentation follows [14, Ch. 3.2] adapted to
the multihomogeneous case.

######  Definition 2.3 (Generic multihomogeneous system).

Consider the set of variables
$\boldsymbol{u}:=\\{u_{k,\boldsymbol{\alpha}}:0\leq k\leq N\text{ and
}\boldsymbol{\alpha}\in\mathcal{A}(\boldsymbol{d}_{k})\\}$ and the ring
$\mathbb{Z}[\boldsymbol{u}]$. The generic multihomogeneous polynomial system
is the system
$\boldsymbol{F}:=\\{F_{0},\dots,F_{N}\\}\subset\mathbb{Z}[\boldsymbol{u}][\boldsymbol{\bar{x}}]$,
where

$F_{k}:=\sum\nolimits_{\boldsymbol{\alpha}\in\mathcal{A}({\boldsymbol{d}_{k}})}u_{k,\boldsymbol{\alpha}}\boldsymbol{x}^{\boldsymbol{\alpha}}.$
(4)

The generic multihomogeneous system $\boldsymbol{F}$ parameterizes every
overdetermined multihomogeneous system with polynomials of multidegrees
${\boldsymbol{d}_{0}},{\boldsymbol{d}_{1}},\dots,{\boldsymbol{d}_{N}}$,
respectively. For each $\boldsymbol{c}=(c_{k,\boldsymbol{\alpha}})_{0\leq
k\leq
N,\boldsymbol{\alpha}\in\mathcal{A}({\boldsymbol{d}_{k}})}\in\mathbb{P}^{\\#\mathcal{A}({\boldsymbol{d}_{0}})-1}\times\dots\times\mathbb{P}^{\\#\mathcal{A}({\boldsymbol{d}_{N}})-1}$,
the specialization of $\boldsymbol{F}$ at $\boldsymbol{c}$, that is
$\boldsymbol{F}(\boldsymbol{c})$, is a multihomogeneous polynomial system in
$\mathbb{K}[\boldsymbol{\bar{x}}]$, say $(f_{0},\dots,f_{N})$, where

$f_{k}:=F_{k}(\boldsymbol{c})=\sum\nolimits_{\boldsymbol{\alpha}\in\mathcal{A}({\boldsymbol{d}_{k}})}c_{k,\boldsymbol{\alpha}}\boldsymbol{x}^{\boldsymbol{\alpha}}.$
(5)

Let $\Omega$ be its incidence variety, that is the algebraic variety
containing the overdetermined multihomogeneous systems that have solutions
over $\mathcal{P}$ and their solutions,

$\Omega=\left\\{(\boldsymbol{p},\boldsymbol{c})\in\mathcal{P}\times(\mathbb{P}^{\\#\mathcal{A}({\boldsymbol{d}_{0}})-1}\times\dots\times\mathbb{P}^{\\#\mathcal{A}({\boldsymbol{d}_{N}})-1}):(\forall
k\in[N])\,F_{k}(\boldsymbol{c})(\boldsymbol{p})=0\right\\}.$

Let $\pi$ be the projection of $\Omega$ to
$\mathbb{P}^{\\#\mathcal{A}({\boldsymbol{d}_{0}})-1}\times\dots\times\mathbb{P}^{\\#\mathcal{A}({\boldsymbol{d}_{N}})-1}$,
that is $\pi(\boldsymbol{p},\boldsymbol{c})=\boldsymbol{c}$. We can think of
$\pi(\Omega)$ as the set of overdetermined multihomogeneous polynomial systems
with solutions over $\mathcal{P}$. This set is an irreducible hypersurface
[28, Prop. 3.3.1]. Its defining ideal in $\mathbb{Z}[\boldsymbol{u}]$ is
principal and it is generated by an irreducible polynomial
$\mathtt{elim}\in\mathbb{Z}[\boldsymbol{u}]$ [28, Prop. 8.1.1]. In particular,
it holds

$\text{The system }\boldsymbol{F}(\boldsymbol{c})\text{ has a solution over
}\mathcal{P}\iff\boldsymbol{c}\in\pi(\Omega)\iff\mathtt{elim}(\boldsymbol{c})=0.$

Following [43, 17, 20], we call $\mathtt{elim}\in\mathbb{Z}[\boldsymbol{u}]$
the eliminant. We warn the reader that the polynomial
$\mathtt{elim}\in\mathbb{Z}[\boldsymbol{u}]$ is called the resultant in [28].
In this work we reserve the word resultant for a power of $\mathtt{elim}$.
More precisely, the resultant $\mathtt{res}$ is a polynomial in
$\mathbb{Z}[\boldsymbol{u}]$ such that
$\mathtt{res}=\pm\mathtt{elim}^{\mathcal{D}},$ where $\mathcal{D}$ is the
degree of the restriction of $\pi$ to the incidence variety $\Omega$, see [20,
Def. 3.1]. Consequently, we have

$\displaystyle\text{The system }\boldsymbol{F}(\boldsymbol{c})\text{ has a
solution over }\mathcal{P}\iff\mathtt{res}(\boldsymbol{c})=0.$ (6)

######  Proposition 2.4 ([43, Prop. 3.4]).

Let ${\boldsymbol{u}_{k}}$ be the blocks of variables in $\boldsymbol{u}$
related to the polynomial $F_{k}$, that is
${\boldsymbol{u}_{k}}=\\{{\boldsymbol{u}_{k,\boldsymbol{\alpha}}}\\}_{\boldsymbol{\alpha}\in\mathcal{A}({\boldsymbol{d}_{i}})}.$
The resultant $\mathtt{res}\in\mathbb{Z}[\boldsymbol{u}]$ is a
multihomogeneous polynomial with respect to the blocks of variables
${\boldsymbol{u}_{0}},\dots,{\boldsymbol{u}_{N}}$. The degree of
$\mathtt{res}$ with respect to the variables ${\boldsymbol{u}_{k}}$ is the
multihomogeneous Bézout bound (Prop. 2.2) of a square system with multidegrees
${\boldsymbol{d}_{0}}\dots\boldsymbol{d}_{k-1},{\boldsymbol{d}_{k+1}}\dots\boldsymbol{d}_{N}$,

$\text{degree}(\mathtt{res},{\boldsymbol{u}_{k}})=\mathtt{MHB}({\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{k-1}},{\boldsymbol{d}_{k+1}},\dots,{\boldsymbol{d}_{N}}).$

The total degree of the resultant is
$\text{degree}(\mathtt{res})=\sum_{k=0}^{N}\mathtt{MHB}({\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{k-1}},{\boldsymbol{d}_{k+1}},\dots,{\boldsymbol{d}_{N}}).$

Usually we compute the resultant as the quotient of the determinants of two
matrices, see [28, Thm. 3.4.2]. When we can compute it as the (exact)
determinant of a matrix, then we say that we have a _determinantal formula_.

###  2.3 Weyman complex

A complex $K_{\bullet}$ is a sequence of free modules
$\\{K_{v}\\}_{v\in\mathbb{Z}}$ together with morphisms
$\delta_{v}:K_{v}\rightarrow K_{v-1}$, such that the image of $\delta_{v}$
belongs to the kernel of $\delta_{v-1}$, that is $(\forall
v\in\mathbb{Z})\;\mathrm{Im}(\delta_{v})\subseteq\mathrm{Ker}(\delta_{v-1})$
or, equivalently, $\delta_{v-1}\circ\delta_{v}=0$. We write $K_{\bullet}$ as

$K_{\bullet}:\cdots\xrightarrow{\delta_{v+1}}K_{v}\xrightarrow{\delta_{v}}K_{v-1}\xrightarrow{\delta_{v-1}}\cdots.$

The complex is exact if for all $v\in\mathbb{Z}$ it holds
$\mathrm{Im}(\delta_{v})=\mathrm{Ker}(\delta_{v-1})$. A complex is bounded
when there are two constants $a$ and $b$ such that, for every $v$ such that
$v<a$ or $b<v$, it holds $K_{v}=0$. If we fix a basis for each $K_{v}$, then
we can represent the maps $\delta_{v}$ using matrices. For a particular class
of bounded complexes, called generically exact (see for example [28, Ap. A]),
we can extend the definition of the determinant of matrices to complexes. The
non-vanishing of the determinant is related to the exactness of the complex.
When there are only two non-zero free modules in the complex (that is all the
other modules are zero) we can define the determinant of the complex if and
only if both the non-zero free modules have the same rank. In this case, the
determinant of the complex reduces to the determinant of the (matrix of the)
map between the two non-zero vector spaces. We refer the reader to [1] for an
accessible introduction to the determinant of a complex and to [28, Ap. A] for
a complete formalization.

The Weyman complex [52, 54, 53] of an overdetermined multihomogeneous system
$\boldsymbol{f}=(f_{0},\dots,f_{N})$ in $\mathbb{K}[\boldsymbol{\bar{x}}]$ is
a bounded complex that is exact if and only if the system $\boldsymbol{f}$ has
no solutions over $\mathcal{P}$. More precisely, the determinant of the Weyman
complex of the multihomogeneous generic system $\boldsymbol{F}$ (see Def. 2.3)
is well-defined and it is equal to the multihomogeneous resultant [53, Prop.
9.1.3]. If the Weyman complex involves only two non-zero vector spaces, then
the resultant of $\boldsymbol{F}$ is the determinant of the map between these
spaces. Thus, in this case, there is a determinantal formula for the
resultant.

######  Theorem 2.5 (Weyman complex, [54, Prop. 2.2]).

Let $\boldsymbol{F}=(F_{0},\dots,F_{N})$ in
$\mathbb{Z}[\boldsymbol{u}][\boldsymbol{\bar{x}}]$ be a generic
multihomogeneous system having multidegrees
${\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{N}}$, respectively (see Def.
2.3). Given a _degree vector_ $\boldsymbol{m}\in\mathbb{Z}^{q}$, there exists
a complex of free $\mathbb{Z}[\boldsymbol{u}]$-modules
$K_{\bullet}(\boldsymbol{m})$, called the _Weyman complex_ of
$\boldsymbol{F}$, such that the determinant of the complex
$K_{\bullet}(\boldsymbol{m})$ agrees with the resultant
$\mathtt{res}(F_{0},\dots,F_{N})$.

$K_{\bullet}(\boldsymbol{m})\\!:\\!0\rightarrow
K_{N+1}(\boldsymbol{m})\xrightarrow{\delta_{N+1}(\boldsymbol{m})}\cdots\rightarrow
K_{1}(\boldsymbol{m})\xrightarrow{\delta_{1}(\boldsymbol{m})}K_{0}(\boldsymbol{m})\xrightarrow{\delta_{0}(\boldsymbol{m})}\cdots\rightarrow
K_{-N}(\boldsymbol{m})\rightarrow 0.$

Moreover, for each $v\in\\{-N,\dots,N+1\\}$ the
$\mathbb{Z}[\boldsymbol{u}]$-module $K_{v}(\boldsymbol{m})$ is

$\displaystyle
K_{v}(\boldsymbol{m})\\!:=\\!\\!\bigoplus_{p=0}^{N+1}K_{v,p}(\boldsymbol{m})\otimes_{\mathbb{Z}}\mathbb{Z}[\boldsymbol{u}],\text{
where
}\,\,\\!K_{v,p}(\boldsymbol{m}):=\\!\\!\\!\\!\\!\\!\\!\bigoplus_{\begin{subarray}{c}I\subset\\{0,\dots,N\\}\\\
\\#I=p\end{subarray}}\\!\\!\\!\\!\\!H^{p-v}_{\mathcal{P}}(\boldsymbol{m}-\sum_{k\in
I}{{\boldsymbol{d}_{k}}})\otimes\\!\\!\bigwedge_{k\in I}e_{k},$ (7)

the term $H^{p-v}_{\mathcal{P}}(\boldsymbol{m}-\sum_{k\in
I}{{\boldsymbol{d}_{k}}})$ is the $(p-v)$-th sheaf cohomology of $\mathcal{P}$
with coefficients in the sheaf
$\mathcal{O}_{\mathcal{P}}(\boldsymbol{m}-\sum_{k\in
I}{{\boldsymbol{d}_{k}}})$ whose global sections are
$\mathbb{K}[\boldsymbol{\bar{x}}]_{\boldsymbol{m}-\sum_{k\in
I}{{\boldsymbol{d}_{k}}}}$ (see [31, Sec. II.5]), 222 The standard notation
for the sheaf cohomology $H^{p}_{\mathcal{P}}(\boldsymbol{m})$, e.g., [31], is
$H^{p}({\mathcal{P}},\mathcal{L}(\sum_{i}m_{i}\,D_{i}))$, where each $D_{i}$
is a Cartier divisor given by the pullback of a hyperplane on
$\mathbb{P}^{n_{i}}$ (via projection) and $\mathcal{L}(\sum_{i}m_{i}\,D_{i})$
is the line bundle associated to the Cartier divisor $(\sum_{i}m_{i}\,D_{i})$
on $\mathcal{P}$. We use our notation for simplicity.  and the element
$\bigwedge_{k\in I}e_{k}$ is the singleton $\\{e_{I_{1}}\wedge\dots\wedge
e_{I_{p}}\\}$, where $I_{1}<\dots<I_{p}$ are the elements of $I$,
$e_{0},\\!\dots,\\!e_{N}$ is the standard basis of $\mathbb{K}^{N+1}$, and
$\wedge$ is the wedge (exterior) product.

For a multihomogeneous system $\boldsymbol{f}=(f_{0},\dots,f_{N})$ in
$\mathbb{K}[\boldsymbol{\bar{x}}]$ that is the specialization of
$\boldsymbol{F}$ at $\boldsymbol{c}$, see (5), the Weyman complex
$K_{\bullet}(\boldsymbol{m};\boldsymbol{f})$ is the Weyman complex
$K_{\bullet}(\boldsymbol{m})$ where we specialize each variable
$u_{k,\boldsymbol{\alpha}}$ at $c_{k,\boldsymbol{\alpha}}\in\mathbb{K}$.

######  Proposition 2.6 ([54, Prop. 2.1]).

The vector spaces $K_{v}(\boldsymbol{m},\boldsymbol{f})$ are independent of
the specialization of the variables $\boldsymbol{u}$, in particular
$K_{v}(\boldsymbol{m},\boldsymbol{f})=\bigoplus_{p=0}^{N+1}K_{v,p}(\boldsymbol{m}).$
Hence, the rank of $K_{v}(\boldsymbol{m})$ as a
$\mathbb{Z}[\boldsymbol{u}]$-module equals the dimension of
$K_{v}(\boldsymbol{m},\boldsymbol{f})$ as a $\mathbb{K}$-vector space. The
differentials $\delta_{v}(\boldsymbol{m},\boldsymbol{f})$ depend on the
coefficients of $\boldsymbol{f}$.

Following [54], as $\mathcal{P}$ is a product of projective spaces, we use
Künneth formula (Prop. 2.7) to write the cohomologies in (7) as a product of
cohomologies of projective spaces, that in turn we can identify with
polynomial rings.

######  Proposition 2.7 (Künneth Formula).

The cohomologies of the product of projective spaces in each
$K_{v,p}(\boldsymbol{m})$ of (7) are the direct sum of the tensor product of
the cohomologies of each projective space, that is

$\displaystyle H^{p-v}_{\mathcal{P}}\Big{(}\boldsymbol{m}-\sum_{k\in
I}{\boldsymbol{d}_{k}}\Big{)}\cong\bigoplus_{r_{1}+\dots+r_{q}=p-v}\bigotimes_{i=1}^{q}H^{r_{i}}_{\mathbb{P}^{n_{i}}}\Big{(}m_{i}-\sum_{k\in
I}d_{k,i}\Big{)}.$ (8)

By combining Bott formula and Serre’s duality, see [41, Sec. 1.1], we can
identify the cohomologies of the previous proposition with the rings
$\mathbb{K}[\boldsymbol{x_{i}}]$ and $\mathbb{K}[\boldsymbol{\partial
x_{i}}]$. Moreover, for each $p-v$, there is at most one set of values for
$(r_{1},\dots,r_{q})$ such that every cohomology in the tensor product of the
right hand side of the previous equation does not vanish. In other words, the
right hand side of (8) reduces to the tensor product of certain cohomologies
of different projective spaces.

######  Remark 2.8.

For each $1\leq i\leq q$, $a\in\mathbb{Z}$, it holds

  * •

$H^{0}_{\mathbb{P}^{n_{i}}}(a)\cong S_{i}(a)$, that is the $\mathbb{K}$-vector
space of the polynomials of degree $a$ in the polynomial algebra
$\mathbb{K}[{\boldsymbol{x}_{i}}]$.

  * •

$H^{n_{i}}_{\mathbb{P}^{n_{i}}}(a)\cong S_{i}^{*}(a+n_{i}+1)$, that is the
$\mathbb{K}$-vector space of the polynomials of degree $a+n_{i}+1$ in the
polynomial algebra $\mathbb{K}[{\boldsymbol{\partial x}_{i}}]$.

  * •

If $r_{i}\not\in\\{0,n_{i}\\}$, then $H^{r_{i}}_{\mathbb{P}^{n_{i}}}(a)\cong
0$.

######  Remark 2.9.

For each $1\leq i\leq q$, if $H^{r_{i}}_{\mathbb{P}^{n_{i}}}(a)\neq 0$, then
$r_{i}\in\\{0,n_{i}\\}$. Moreover,

  * •

If $a>-n_{i}-1$, then $\displaystyle H^{r_{i}}_{\mathbb{P}^{n_{i}}}(a)\neq
0\iff r_{i}=0\text{ and }a\geq 0.$

  * •

If $a<0$, then $\displaystyle\qquad H^{r_{i}}_{\mathbb{P}^{n_{i}}}(a)\neq
0\iff r_{i}=n_{i}\text{ and }a\leq-n_{i}-1.$

We define the dual of a complex by dualizing the modules and the maps. The
dual of the Weyman complex is isomorphic to another Weyman complex. By
exploiting Serre’s duality, we can construct the degree vectors of a dual
Weyman complex from the degree vector of the primal.

######  Proposition 2.10 ([53, Thm. 5.1.4]).

Let $\boldsymbol{m}$ and $\boldsymbol{\bar{m}}$ be any degree vectors such
that
$\boldsymbol{m}+\boldsymbol{\bar{m}}=\sum_{i}\boldsymbol{d}_{i}-(n_{1}+1,\dots,n_{q}+1)$.
Then, $K_{v}(\boldsymbol{m})\cong K_{1-v}(\boldsymbol{\bar{m}})^{*}$ for all
$v\in{\mathbb{Z}}$ and $K_{\bullet}(\boldsymbol{m})$ is dual to
$K_{\bullet}(\boldsymbol{\bar{m}})$.

###  2.4 Koszul-type formula

Our goal is to obtain determinantal formulas given by matrices whose elements
are linear forms in the coefficients of the input polynomials, that is linear
in $\boldsymbol{u}$, see (4). Hence, by [53, Prop. 5.2.4], our objective is to
choose a degree vector $\boldsymbol{m}$ so that the Weyman complex reduces to

$\displaystyle K_{\bullet}(\boldsymbol{m}):0\rightarrow
K_{v,p+v}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\xrightarrow{\delta_{v}(\boldsymbol{m})}K_{v-1,p+v-1}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\rightarrow
0\,,$ (9)

where $p=\sum_{k\in I}n_{k}$, for some set $I\subset\\{1,\dots,q\\}$. That is,
it holds
$K_{v}(\boldsymbol{m})=K_{v,p+v}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]$,
$K_{v-1}(\boldsymbol{m})=K_{v-1,p+v-1}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]$,
and, for all $t\not\in\\{v-1,v\\}$, $K_{t}(\boldsymbol{m})=0$,

We will describe the map $\delta_{v}(\boldsymbol{m})$ through an auxiliary map
$\mu$ that acts as multiplication. For this, we need to introduce some
additional notation. This notation is independent from the rest of the paper
and the readers that are familiar with the Weyman complex can safely skip the
rest of the section. Let $R$ be a ring; for example
$R=\mathbb{Z}[\boldsymbol{u}]$ or $R=\mathbb{K}$. For each $1\leq i\leq q$,
the polynomial ring $R[{\boldsymbol{x}_{i}}]$, respectively
$R[{\boldsymbol{\partial x}_{i}}]$, is a free $R$-module with basis
$\\{\boldsymbol{x}_{i}^{\boldsymbol{\alpha}}:\boldsymbol{\alpha}\in\mathcal{A}(d),d\in\mathbb{Z}\\}$,
respectively $\\{\boldsymbol{\partial
x}_{i}^{\boldsymbol{\alpha}}:\boldsymbol{\alpha}\in\mathcal{A}(d),d\in\mathbb{Z}\\}$.
We define the bilinear map

$\displaystyle\mu_{(i)}:R[{\boldsymbol{x}_{i}}]\times(R[{\boldsymbol{x}_{i}}]\oplus
R[{\boldsymbol{\partial x}_{i}}])\rightarrow R[{\boldsymbol{x}_{i}}]\oplus
R[{\boldsymbol{\partial x}_{i}}],$ (10)

which acts follows: for each $d_{1},d_{2}\in\mathbb{Z}$,
$\boldsymbol{\alpha}\in\mathcal{A}(d_{1})$ and
$\boldsymbol{\beta},\boldsymbol{\gamma}\in\mathcal{A}(d_{2})$, we have

$\displaystyle\mu_{(i)}(\boldsymbol{x}_{i}^{\boldsymbol{\alpha}},\boldsymbol{x}_{i}^{\boldsymbol{\gamma}})=\;\boldsymbol{x}_{i}^{{\boldsymbol{\alpha}}+{\boldsymbol{\gamma}}}\,\text{
and }\,\mu_{(i)}(\boldsymbol{x}_{i}^{\boldsymbol{\alpha}},\boldsymbol{\partial
x}_{i}^{\boldsymbol{\beta}})=$ $\displaystyle\left\\{\begin{array}[]{l
l}\boldsymbol{\partial
x}_{i}^{{\boldsymbol{\beta}}-{\boldsymbol{\alpha}}}&\text{if $d_{1}\leq d_{2}$
and
${\boldsymbol{\beta}}-{\boldsymbol{\alpha}}\in\mathcal{A}(d_{1}-d_{2})$}\\!\\!\\!\\!\\!\\!\\\
0&\text{otherwise}.\end{array}\right.$

The map $\mu_{(i)}$ is graded in the following way, for $f\in S_{i}(d)$ it
holds

$\mu_{(i)}(f,S_{i}(D))\subseteq
S_{i}(D+d)\quad\text{and}\quad\mu_{(i)}(f,S_{i}^{*}(D))\subseteq
S_{i}^{*}(D+d).$

We define the bilinear map $\mu:=\bigotimes_{i=1}^{q}\mu_{(i)}$.

######  Remark 2.11.

If we restrict the domain of $\mu$ to
$\bigotimes_{i=1}^{q}\left(R[{\boldsymbol{x}_{i}}]\times
R[{\boldsymbol{x}_{i}}]\right)\simeq\left(\bigotimes_{i=1}^{q}R[{\boldsymbol{x}_{i}}]\right)\times\left(\bigotimes_{i=1}^{q}R[{\boldsymbol{x}_{i}}]\right)$,
then $\mu$ acts as multiplication, ie for
$f,g\in\left(\bigotimes_{i=1}^{q}R[{\boldsymbol{x}_{i}}]\right)$, it holds
$\mu(f,g)=f\,g.$

Given $f\in\left(\bigotimes_{i=1}^{q}R[{\boldsymbol{x}_{i}}]\right)$, we
define the linear map

$\displaystyle\begin{array}[]{r r
c}\mu_{f}:&\bigotimes_{i=1}^{q}(R[{\boldsymbol{x}_{i}}]\oplus
R[\boldsymbol{\partial
x_{i}}])\rightarrow&\bigotimes_{i=1}^{q}(R[{\boldsymbol{x}_{i}}]\oplus
R[{\boldsymbol{\partial x}_{i}}])\\\
&g\mapsto&\mu_{f}(g)=\mu(f,g).\end{array}$

Using the isomorphisms of Prop. 2.7 and Rem. 2.8, for
$\boldsymbol{d}\in\mathbb{N}^{q}$ and
$f\in\mathbb{K}[\boldsymbol{\bar{x}}]_{\boldsymbol{d}}$, if we restrict the
map $\mu_{f}$ to $H^{r}_{\mathcal{P}}(\boldsymbol{m})$, for any
$r\in\mathbb{N}$, then we obtain the map
$\mu_{f}:H^{r}_{\mathcal{P}}(\boldsymbol{m})\rightarrow
H^{r}_{\mathcal{P}}(\boldsymbol{m}+\boldsymbol{d}).$

######  Definition 2.12 (Inner derivative [54]).

 Let $E$ be a $\mathbb{K}$-vector space generated by  $\\{e_{1},\dots
e_{N}\\}$. We define the $k$-th inner derivative, $\Delta_{k}$, of the
exterior algebra of $\bigwedge E$ as the $(-1)$-graded map such that, for each
$i$ and $1\leq j_{1}<\dots<j_{i}\leq N$,

$\displaystyle\small\begin{array}[]{r c
l}\Delta_{k}:&\bigwedge^{i}E&\\!\\!\\!\\!\\!\rightarrow\bigwedge^{i-1}E\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!e_{j_{1}}\wedge\dots\wedge
e_{j_{i}}&\\!\\!\\!\\!\\!\mapsto\Delta_{k}(e_{j_{1}}\wedge\dots\wedge
e_{j_{i}})=\left\\{\begin{array}[]{c l}(-1)^{t+1}e_{j_{1}}\wedge\dots\wedge
e_{j_{t-1}}\wedge e_{j_{t+1}}\wedge\dots\wedge e_{j_{i}}\\!\\!\\!&\text{if
}j_{t}=k\\\ 0&\text{otherwise}\end{array}\right.\end{array}$

Given $R$-modules $A_{1},A_{2},B_{1},B_{2}$ and homomorphisms
$\mu_{1}:A_{1}\rightarrow B_{1}$ and $\mu_{2}:A_{2}\rightarrow B_{2}$, their
tensor product is the map $\mu_{1}\otimes\mu_{2}:A_{1}\otimes A_{2}\rightarrow
B_{1}\otimes B_{2}$ such that, for $a_{1}\in A_{1}$ and $a_{2}\in A_{2}$,
$(\mu_{1}\otimes\mu_{2})(a_{1}\otimes
a_{2})=\mu_{1}(a_{1})\otimes\mu_{2}(a_{2}).$

######  Proposition 2.13 ([54, Prop. 2.6]).

Consider the generic overdetermined multihomogeneous system
$\boldsymbol{F}\in\mathbb{Z}[\boldsymbol{u}][\boldsymbol{\bar{x}}]^{N+1}$ with
polynomials of multidegrees ${\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{N}}$,
respectively (Def. 2.3). Given a degree vector
$\boldsymbol{m}\in\mathbb{Z}^{q}$, we consider the Weyman complex
$K_{\bullet}(\boldsymbol{m})$. If there is $v\in\\{-N+1,\dots,N+1\\}$ and
$p\in\\{0,\dots,N+1\\}$ such that

$K_{v}(\boldsymbol{m})=K_{v,p}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\qquad\text{and}\qquad
K_{v-1}(\boldsymbol{m})=K_{v-1,p-1}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}],$

then the map $\delta_{v}(\boldsymbol{m}):K_{v}(\boldsymbol{m})\rightarrow
K_{v-1}(\boldsymbol{m})$ is
$\delta_{v}(\boldsymbol{m})=\sum_{k=0}^{N}\mu_{F_{k}}\otimes\Delta_{k},$ where
$\mu_{F_{k}}\otimes\Delta_{k}$ denotes the tensor product of the maps
$\mu_{F_{k}}$ and $\Delta_{k}$ (Def. 2.12).

######  Definition 2.14 (Koszul-type determinantal formula).

With the notation of Prop. 2.13, when the Weyman complex reduces to

$\displaystyle K_{\bullet}(\boldsymbol{m}):0\rightarrow
K_{1,p+1}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\xrightarrow{\delta_{1}(\boldsymbol{m})}K_{0,p}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\rightarrow
0\,,$ (11)

we say that the map $\delta_{1}(\boldsymbol{m})$ is a _Koszul-type
determinantal formula_.

######  Example 2.15.

Consider the blocks of variables $\boldsymbol{x}_{1}:=\\{x_{1,0},x_{1,1}\\}$
and $\boldsymbol{x}_{2}\\!:=\\!\\{x_{2,0},x_{2,1}\\}$, and the systems
$\boldsymbol{f}:=(f_{0},f_{1},f_{2})$ of multidegrees
$\boldsymbol{d}_{0}=\boldsymbol{d}_{1}=\boldsymbol{d}_{2}=(1,1)$. That is,

$\displaystyle\begin{cases}f_{0}=(a_{0,0}\,x_{1,0}+a_{1,0}\,x_{1,1})\,x_{2,0}+(a_{0,1}\,x_{1,0}+a_{1,1}\,x_{1,1})\,x_{2,1}\\\
f_{1}=(b_{0,0}\,x_{1,0}+b_{1,0}\,x_{1,1})\,x_{2,0}+(b_{0,1}\,x_{1,0}+b_{1,1}\,x_{1,1})\,x_{2,1}\\\
f_{2}=(c_{0,0}\,x_{1,0}+c_{1,0}\,x_{1,1})\,x_{2,0}+(c_{0,1}\,x_{1,0}+c_{1,1}\,x_{1,1})\,x_{2,1}.\end{cases}$
(12)

As in [23, Lem. 2.2], consider the degree vector $\boldsymbol{m}=(2,-1)$. So,
the Weyman complex is

$K_{\bullet}(\boldsymbol{m},\boldsymbol{f}):0\rightarrow
K_{1,2}(\boldsymbol{m},\boldsymbol{f})\xrightarrow{\delta_{1}(\boldsymbol{m},\boldsymbol{f})}K_{0,1}(\boldsymbol{m},\boldsymbol{f})\rightarrow
0,$

where

$\displaystyle\left\\{\begin{array}[]{l
c}K_{1,2}(\boldsymbol{m},\boldsymbol{f})=&S_{1}(0)\otimes
S_{2}^{*}(-1)\otimes\Big{(}\\{e_{0}\wedge e_{1}\\}\oplus\\{e_{0}\wedge
e_{2}\\}\oplus\\{e_{1}\wedge e_{2}\\}\Big{)}\\\
K_{0,1}(\boldsymbol{m},\boldsymbol{f})=&S_{1}(1)\otimes
S_{2}^{*}(0)\otimes\Big{(}\\{e_{0}\\}\oplus\\{e_{1}\\}\oplus\\{e_{2}\\}\Big{)}.\end{array}\right.$

If we consider monomial bases for $K_{1,2}(\boldsymbol{m})$ and
$K_{0,1}(\boldsymbol{m})$, then we can represent $\delta_{1}(\boldsymbol{m})$
with the transpose of the matrix that follows. Note that, the element
$\partial 1\in\mathbb{K}[\boldsymbol{\partial x}_{1},\boldsymbol{\partial
x}_{1}]$ corresponds to the dual of
$1\in\mathbb{K}[{\boldsymbol{x}_{1}},{\boldsymbol{x}_{2}}]$.

$\displaystyle\begin{array}[]{c | c c c c c c}&x_{1,0}\otimes\partial 1\otimes e_{0}&&x_{1,0}\otimes\partial 1\otimes e_{1}&&x_{1,0}\otimes\partial 1\otimes e_{2}&\\\ &&x_{1,1}\otimes\partial 1\otimes e_{0}&&x_{1,1}\otimes\partial 1\otimes e_{1}&&x_{1,1}\otimes\partial 1\otimes e_{2}\\\ \hline\cr 1\otimes{\partial x}_{2,0}\otimes(e_{0}\wedge e_{1})&-b_{0,0}&-b_{1,0}&a_{0,0}&a_{1,0}&0&0\\\ 1\otimes{\partial x}_{2,1}\otimes(e_{0}\wedge e_{1})&-b_{0,1}&-b_{1,1}&a_{0,1}&a_{1,1}&0&0\\\ 1\otimes{\partial x}_{2,0}\otimes(e_{0}\wedge e_{2})&-c_{0,0}&-c_{1,0}&0&0&a_{0,0}&a_{1,0}\\\ 1\otimes{\partial x}_{2,1}\otimes(e_{0}\wedge e_{2})&-c_{0,1}&-c_{1,1}&0&0&a_{0,1}&a_{1,1}\\\ 1\otimes{\partial x}_{2,0}\otimes(e_{1}\wedge e_{2})&0&0&-c_{0,0}&-c_{1,0}&b_{0,0}&b_{1,0}\\\ 1\otimes{\partial x}_{2,1}\otimes(e_{1}\wedge e_{2})&0&0&-c_{0,1}&-c_{1,1}&b_{0,1}&b_{1,1}\\\ \end{array}$

The resultant is equal (up to sign) to the determinant of the above matrix, it
has total degree 6 (same as the size of this matrix) and 66 terms.

As we saw in the previous example, once we have fixed a basis for the map in
Prop. 2.13, we can represent the Koszul-type determinantal formula by the
determinant of a matrix. We refer to this matrix as a _Koszul resultant
matrix_.

######  Corollary 2.16 ([53, Prop. 5.2.4]).

Let $\boldsymbol{F}$ be a generic multihomogeneous system of polynomials with
multidegrees ${\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{N}}$, respectively.
Let $\boldsymbol{m}\in\mathbb{Z}^{q}$ be a degree vector so that the Weyman
complex $K_{\bullet}(\boldsymbol{m})$ becomes

$\displaystyle K_{\bullet}(\boldsymbol{m}):0\rightarrow
K_{v,p+v}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\xrightarrow{\delta_{v}(\boldsymbol{m})}K_{v-1,p+v-1}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\rightarrow
0\,.$

Then, the map $\delta_{v}(\boldsymbol{m})$ of Prop. 2.13 is linear in the
coefficients of $\boldsymbol{F}$. Moreover, as the determinant of the complex
is the resultant, the rank of both $K_{v}(\boldsymbol{m})$ and
$K_{v+1}(\boldsymbol{m})$, as $\mathbb{Z}[\boldsymbol{u}]$-modules, equals the
degree of the resultant (Prop. 2.4).

We remark that Koszul-type formulas generalizes Sylvester-type formulas.

######  Proposition 2.17.

Under the assumptions of Prop. 2.13, if $p=1$ and $v=0$, the map
$\delta_{v}(\boldsymbol{m})$ acts as a Sylvester map, that is
$(g_{0},\dots,g_{N})\mapsto\sum_{k=0}^{N}g_{k}\,F_{k}$. In this case, it holds

$\delta_{v}(\boldsymbol{m})(g_{0}\otimes e_{0}+\dots+g_{N}\otimes
e_{N})=\Big{(}\sum_{k=0}^{N}g_{k}\,F_{k}\Big{)}\otimes 1.$

Determinantal formulas for the multiprojective resultant of unmixed systems,
that is systems where the multidegree of each polynomial is the same, were
extensively studied by several authors [47, 54, 12, 18]. However, there are
very few results about determinantal formulas for mixed multihomogeneous
systems, that is, when the supports are not the same. We know such formulas
for scaled multihomogeneous systems [22], that is when the supports are scaled
copies of one of them, and for bivariate tensor-product polynomial systems
[38, 9]. In what follows, we use the Weyman complex to derive new formulas for
families of mixed multilinear systems.

##  3 Determinantal formulas for star multilinear systems

We consider four different kinds of overdetermined multihomogeneous systems,
related to _star multilinear systems_ (Def. 3.1) and we construct
determinantal formulas for each of them. These formulas are Koszul- and
Sylvester-type determinantal formulas (Def. 2.14). To simplify the
presentation, we change somewhat the notation that we used for the polynomial
systems in sec. 2.1. We split the blocks of variables in two groups; we
replace the blocks of variables $\boldsymbol{x}_{k}$ by ${\boldsymbol{X}_{i}}$
or ${\boldsymbol{Y}_{j}}$ and the constants $n_{k}$, that correspond to the
cardinalities of the blocks, by $\alpha_{i}$ or $\beta_{j}$. Let
$A,B\in\mathbb{N}$ and $q=A+B$. Let $\boldsymbol{\bar{X}}$ be the set of $A$
blocks of variables $\\{\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{A}\\}$. For
each $i\in[A]$, ${\boldsymbol{X}_{i}}:=\\{x_{i,0},\dots,x_{i,\alpha_{i}}\\}$;
so the number of affine variables in each block $\boldsymbol{X}_{i}$ is
$\alpha_{i}\in{\mathbb{N}}$. We also consider the polynomial algebra
$\mathbb{K}[{\boldsymbol{X}_{i}}]=\bigoplus_{d\in\mathbb{Z}}S_{\\!\boldsymbol{X}_{i}}(d)$,
where $S_{\\!\boldsymbol{X}_{i}}(d)$ is the $\mathbb{K}$-vector space of
polynomials of degree $d$ in $\mathbb{K}[{\boldsymbol{X}_{i}}]$. Similarly,
$\boldsymbol{\bar{Y}}$ is the set of $B$ blocks of variables
$\\{\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{B}\\}$. For each $j\in[B]$,
${\boldsymbol{Y}_{j}}:=\\{y_{j,0},\dots,y_{j,\beta_{j}}\\}$; hence the number
of variables in each block ${\boldsymbol{Y}_{j}}$ is
$\beta_{j}\in{\mathbb{N}}$. Moreover,
$\mathbb{K}[{\boldsymbol{Y}_{j}}]=\bigoplus_{d\in\mathbb{Z}}S_{\boldsymbol{Y}_{j}}(d)$,
where $S_{\boldsymbol{Y}_{j}}(d)$ is the $\mathbb{K}$-vector space of
polynomials of degree $d$ in $\mathbb{K}[{\boldsymbol{Y}_{j}}]$.

Consider the $\mathbb{Z}^{q}$-multigraded algebra
$\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$, given by

$\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]:=\\!\\!\\!\\!\\!\\!\\!\\!\bigoplus_{(d_{\boldsymbol{X}_{1}},\dots,d_{\boldsymbol{X}_{A}},d_{\boldsymbol{Y}_{1}},\dots,d_{\boldsymbol{Y}_{B}})\in\mathbb{Z}^{q}}\\!\\!\\!\\!\\!\\!\\!\\!S_{\boldsymbol{X}_{1}}(d_{\boldsymbol{X}_{1}})\otimes\dots\otimes
S_{\boldsymbol{X}_{A}}(d_{\boldsymbol{X}_{A}})\otimes
S_{\boldsymbol{Y}_{1}}(d_{\boldsymbol{Y}_{1}})\otimes\dots\otimes
S_{\boldsymbol{Y}_{B}}(d_{\boldsymbol{Y}_{B}}).$

For a multihomogeneous polynomial
$f\in\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$ of multidegree
$\boldsymbol{d}\in\mathbb{Z}^{q}$, we denote by $d_{\boldsymbol{X}_{i}}$,
respectively $d_{\boldsymbol{Y}_{j}}$, the degree of $f$ with respect to the
block of variables ${\boldsymbol{X}_{i}}$, respectively
${\boldsymbol{Y}_{j}}$. For each group of indices $1\leq i_{1}<\dots<i_{r}\leq
A$ and $1\leq j_{1}<\dots<j_{s}\leq B$, we denote by
$\mathbb{K}[{\boldsymbol{X}_{i_{1}}},\dots,{\boldsymbol{X}_{i_{r}}},{\boldsymbol{Y}_{j_{1}}},\dots,{\boldsymbol{Y}_{j_{s}}}]_{\boldsymbol{1}}$
the set of multilinear polynomials in
$\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$ with multidegree
$(d_{\boldsymbol{X}_{1}},\dots,d_{\boldsymbol{X}_{A}},d_{\boldsymbol{Y}_{1}},\dots,d_{\boldsymbol{Y}_{B}})$,
where

$\displaystyle d_{\boldsymbol{X}_{l}}=\left\\{\begin{array}[]{c l}1&\text{if
}l\in\\{i_{1},\dots,i_{r}\\}\\\
0&\text{otherwise}\end{array}\right.\quad\text{ and }$ $\displaystyle\quad
d_{\boldsymbol{Y}_{l}}=\left\\{\begin{array}[]{c l}1&\text{if
}l\in\\{j_{1},\dots,j_{s}\\}\\\ 0&\text{otherwise}.\end{array}\right..$

Let $N=\sum_{i=1}^{A}\alpha_{i}+\sum_{j=1}^{B}\beta_{j}$. We say that a
polynomial system is _square_ if it has $N$ equations and _overdetermined_ if
it has $N+1$. We consider the multiprojective space

$\mathcal{P}:=\mathbb{P}^{\alpha_{1}}\times\dots\times\mathbb{P}^{\alpha_{A}}\times\mathbb{P}^{\beta_{1}}\times\dots\times\mathbb{P}^{\beta_{B}}.$

######  Definition 3.1 (Star multilinear systems).

A square multihomogeneous system $\boldsymbol{f}=(f_{1},\dots,f_{N})$ in
$\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$ with multidegrees
${\boldsymbol{d}_{1}},\dots,{\boldsymbol{d}_{N}}\in\mathbb{Z}^{q}$,
respectively, is a _Star_ _multilinear system_ if for every $k\in[N]$, there
is $j_{k}\in[B]$ such that

$f_{k}\in\mathbb{K}[{\boldsymbol{X}_{1}},\dots,{\boldsymbol{X}_{A}},{\boldsymbol{Y}_{j_{k}}}]_{\boldsymbol{1}}.$

For each $j\in[B]$, we denote by $\mathcal{E}_{j}$ the number of polynomials
of $\boldsymbol{f}$ in
$\mathbb{K}[{\boldsymbol{X}_{1}},\dots,{\boldsymbol{X}_{A}},{\boldsymbol{Y}_{j}}]_{\boldsymbol{1}}.$

We use the term _star_ because we can represent such systems using a star
graph with weighted edges. The vertices of the graph are the algebras
$\mathbb{K}[{\boldsymbol{Y}_{1}}],\dots,\mathbb{K}[{\boldsymbol{Y}_{B}}]$, and
$\mathbb{K}[{\boldsymbol{X}_{1}},\dots,{\boldsymbol{X}_{A}}]$. For each
${\boldsymbol{d}_{k}}$ there is an edge between the vertices
$\mathbb{K}[{\boldsymbol{X}_{1}},\dots,{\boldsymbol{X}_{A}}]$ and
$\mathbb{K}[{\boldsymbol{Y}_{j}}]$ whenever $d_{k,Y_{j}}=1$. The weight of the
edge between the vertices
$\mathbb{K}[{\boldsymbol{X}_{1}},\dots,{\boldsymbol{X}_{A}}]$ and
$\mathbb{K}[{\boldsymbol{Y}_{j}}]$ corresponds to $\mathcal{E}_{j}$. That is,
when it holds
$f_{k}\in\mathbb{K}[{\boldsymbol{X}_{1}},\dots,{\boldsymbol{X}_{A}},{\boldsymbol{Y}_{j}}]_{\boldsymbol{1}}$.
The graph is a star because every vertex is connected to
$\mathbb{K}[{\boldsymbol{X}_{1}},\dots,{\boldsymbol{X}_{A}}]$ and there is no
edge between two vertices $\mathbb{K}[{\boldsymbol{Y}_{j_{1}}}]$ and
$\mathbb{K}[{\boldsymbol{Y}_{j_{2}}}]$.

######  Example 3.2.

Let
${\boldsymbol{X}_{1}},{\boldsymbol{X}_{2}},{\boldsymbol{Y}_{1}},{\boldsymbol{Y}_{2}},{\boldsymbol{Y}_{3}}$
be blocks of variables. Consider the multihomogeneous system
$(f_{1},f_{2},f_{3},f_{4})\subset\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$
with the following (pattern of) multidegrees

$\displaystyle\begin{array}[]{r c c | c c c c}(\\!\\!\\!\\!&d_{k,X_{1}},&d_{k,X_{2}},&d_{k,Y_{1}},&d_{k,Y_{2}},&d_{k,Y_{3}}&\\!\\!\\!\\!)\\\ {\boldsymbol{d}_{1}}=(\\!\\!\\!\\!&1,&1,&1,&0,&0&\\!\\!\\!\\!)\\\ {\boldsymbol{d}_{2}}=(\\!\\!\\!\\!&1,&1,&1,&0,&0&\\!\\!\\!\\!)\\\ {\boldsymbol{d}_{3}}=(\\!\\!\\!\\!&1,&1,&0,&1,&0&\\!\\!\\!\\!)\\\ {\boldsymbol{d}_{4}}=(\\!\\!\\!\\!&1,&1,&0,&0,&1&\\!\\!\\!\\!)\end{array}$ ${\mathbb{K}[{\boldsymbol{X}_{1}},{\boldsymbol{X}_{2}}]}$${\mathbb{K}[\boldsymbol{Y}_{1}]}$${\mathbb{K}[\boldsymbol{Y}_{2}]}$${\mathbb{K}[\boldsymbol{Y}_{3}]}$$\mathcal{E}_{1}=2$$\mathcal{E}_{2}=1$$\mathcal{E}_{3}=1$

It is a star multilinear system where $\mathcal{E}_{1}=2$,
$\mathcal{E}_{2}=1$, and $\mathcal{E}_{3}=1$. The corresponding star graph is
the one above.

######  Remark 3.3.

For each square star multilinear system, it holds
$N=\sum_{j=1}^{B}\mathcal{E}_{j}$. Moreover, if the system has a nonzero
finite number of solutions, then for each $j\in\\{1,\dots,B\\}$ it holds
$\mathcal{E}_{j}\geq\beta_{j}$, see Prop. 2.2.

###  3.1 Determinantal formulas

To be able to define the resultant, we study overdetermined polynomial systems
$(f_{0},f_{1},\dots,f_{N})$ in $\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$
where $(f_{1},\dots,f_{N})$ is a square star multilinear system and $f_{0}$ is
a multilinear polynomial. The obvious choice for $f_{0}$ is to have the same
structure as one of the polynomials $f_{1},\dots,f_{N}$; still we also choose
$f_{0}$ to have a different support. This leads to resultants of smaller
degrees and so to matrices of smaller size. Aiming at resultant formulas for
_any_ $A$ and $B$, we were able to identify the following choices of $f_{0}$,
that lead to a determinantal Weyman complex333 For other choices of $f_{0}$ we
found specific values of $A$ and $B$ for which every possible Weyman complex
is not determinantal. . In particular, the following $f_{0}$ lead to
determinantal formulas:

  * •

Center-Vertex case:
$f_{0}\in\mathbb{K}[\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{A}]_{\boldsymbol{1}}$.

  * •

Outer-Vertex case: $f_{0}\in\mathbb{K}[\boldsymbol{Y}_{j}]_{1}$, for any
$j\in[B]$.

  * •

Edge case:
$f_{0}\in\mathbb{K}[\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{A},\boldsymbol{Y}_{j}]_{\boldsymbol{1}}$,
for any $j\in[B]$,

  * •

Triangle case:
$f_{0}\in\mathbb{K}[\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{A},\boldsymbol{Y}_{j_{1}},\boldsymbol{Y}_{j_{2}}]_{\boldsymbol{1}}$,
for any $j_{1},j_{2}\in[B]$, $j_{1}\neq j_{2}$.

We can view the various multidegrees of $f_{0}$,
${\boldsymbol{d}_{0}}=(d_{0,{X}_{1}},\dots,d_{0,{X}_{A}},d_{0,{Y_{1}}},\dots,d_{0,{Y_{B}}})$,
in the cases above as solutions of the following system of inequalities:

$\displaystyle\begin{cases}(\forall\,1\leq i\leq A)\,&0\leq
d_{0,\boldsymbol{X}_{i}}\leq 1,\\\ (\forall\,1\leq j\leq B)\,&0\leq
d_{0,\boldsymbol{Y}_{j}}\leq 1,\\\ (\forall\,1\leq i_{1}<i_{2}\leq
A)&d_{0,\boldsymbol{X}_{i_{1}}}=d_{0,\boldsymbol{X}_{i_{2}}}\text{, and }\\\
&\sum_{j=1}^{B}d_{0,\boldsymbol{Y}_{j}}\leq
1+d_{0,\boldsymbol{X}_{1}}.\end{cases}$ (13)

Consider the set $\\{0,\dots,N\\}$ that corresponds to generic polynomials
$\boldsymbol{F}=(F_{0},\dots,F_{N})$ (Def. 2.3). As many of the polynomials
have the same support, we can gather them to simplify the cohomologies of (7).
We need the following notation. For each tuple
$(s_{0},\dots,s_{B})\in\mathbb{N}^{B+1}$, let
$\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}$ be the set of all the subsets of
$\\{0,\dots,N\\}$, such that

  * •

For $1\leq j\leq B$, the index $s_{j}$ indicates that we consider exactly
$s_{j}$ polynomials from $(F_{1},\dots,F_{N})$ that belong to
$\mathbb{Z}[\boldsymbol{u}][\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{A},\boldsymbol{Y}_{j}]_{\boldsymbol{1}}$.

  * •

In addition, if $s_{0}=1$, then 0 belongs to all the sets in
$\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}$.

That is,

$\displaystyle\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}:=\Big{\\{}I:\;$
$\displaystyle I\subset\\{0,\dots,N\\},\left(0\in
I\\!\Leftrightarrow\\!s_{0}=1\right)\text{ and }$ $\displaystyle(\forall 1\leq
j\leq B)\;s_{j}=\\#\\{k\in
I\setminus\\{0\\}:F_{k}\in\mathbb{Z}[\boldsymbol{u}][\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{A},\boldsymbol{Y}_{j}]_{\boldsymbol{1}}\\}\Big{\\}}.$
(14)

######  Example 3.4.

If we consider a system $(F_{1},\dots,F_{4})$ as in Ex. 3.2 and introduce some
$F_{0}$, it holds for $\boldsymbol{F}=(F_{0},\dots,F_{4})$ that
$\mathcal{I}_{1,1,1,0}=\\{\\{0,1,3\\},\\{0,2,3\\}\\}$ and
$\mathcal{I}_{0,2,0,1}=\\{\\{1,2,4\\}\\}$.

Notice that if $I,J\in\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}$, then $I$ and $J$
have the same cardinality and $\sum_{k\in I}{\boldsymbol{d}_{k}}=\sum_{k\in
J}{\boldsymbol{d}_{k}}$, as they correspond to subsets of polynomials of
$\boldsymbol{F}$ with the same multidegrees.

The following lemma uses the sets $\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}$ to
simplify the cohomologies of (7).

######  Lemma 3.5.

Consider a generic overdetermined system
$\boldsymbol{F}\\!=\\!(F_{0},\dots,F_{N})$ in
$\mathbb{Z}[\boldsymbol{u}][\boldsymbol{\bar{X},\bar{Y}}]$ of multidegrees
${\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{N}}$ (Def. 2.3), where
$(F_{1},\dots,F_{n})$ is a square star multilinear system such that, for every
$j\in\\{1,\dots,B\\}$, $\mathcal{E}_{j}\geq\beta_{j}$, and
${\boldsymbol{d}_{0}}$ is the multidegree of $F_{0}$. Following (7), we can
rewrite the modules of the Weyman complex
$K_{v}(\boldsymbol{m})=\bigoplus_{p=0}^{N+1}K_{v,p}\otimes\mathbb{Z}[\boldsymbol{u}]$
in the more detailed form

$\displaystyle
K_{v,p}(\boldsymbol{m})\cong\\!\\!\\!\\!\\!\\!\\!\\!\\!\bigoplus_{\begin{subarray}{c}0\leq
s_{0}\leq 1\\\ 0\leq s_{1}\leq\mathcal{E}_{1}\\\ \dots\\\ 0\leq
s_{B}\leq\mathcal{E}_{B}\\\
s_{0}+s_{1}+\dots+s_{B}=p\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H^{p-v}_{\mathcal{P}}\Big{(}\boldsymbol{m}-(\sum_{j=1}^{B}s_{j},\dots,\sum_{j=1}^{B}s_{j},s_{1},\dots,s_{B})-s_{0}\,{\boldsymbol{d}_{0}}\Big{)}\otimes\\!\\!\\!\bigoplus_{I\in\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}}\bigwedge_{k\in
I}e_{k}.$

Moreover, the following isomorphisms hold for the cohomologies:

$H^{p-v}_{\mathcal{P}}\Big{(}\boldsymbol{m}-(\sum_{j=1}^{B}s_{j},\dots,\sum_{j=1}^{B}s_{j},s_{1},\dots,s_{B})-s_{0}\,{\boldsymbol{d}_{0}}\Big{)}\cong\\\
\bigoplus_{\begin{subarray}{c}\\\ \\\ r_{{\boldsymbol{X}_{1}}}\dots
r_{{\boldsymbol{X}_{1}}},r_{\boldsymbol{Y}_{1}}\dots
r_{\boldsymbol{Y}_{B}}\in\mathbb{N}\\\
\sum_{i}r_{{\boldsymbol{X}_{i}}}+\sum_{j}r_{\boldsymbol{Y}_{j}}=p-v\end{subarray}}\\!\\!\\!\Big{(}\bigotimes_{i=1}^{A}H^{r_{{\boldsymbol{X}_{i}}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}m_{\boldsymbol{X}_{i}}-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,\boldsymbol{X}_{i}}\Big{)}\displaystyle\otimes\bigotimes_{j=1}^{B}H^{r_{\boldsymbol{Y}_{j}}}_{\mathbb{P}^{\beta_{j}}}(m_{{\boldsymbol{Y}_{j}}}-s_{j}-s_{0}\,d_{0,{\boldsymbol{Y}_{j}}})\Big{)}.$
(16)

###### Proof.

Consider $I,J\subset\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}$. Then, by
definition, $\\#I=\\#J$ and $\sum\limits_{k\in
I}{\boldsymbol{d}_{k}}=\sum\limits_{k\in J}{\boldsymbol{d}_{k}}=$
$(\sum_{j=1}^{B}s_{j},\dots,\sum_{j=1}^{B}s_{j},s_{1},\dots,s_{B})+s_{0}\,{\boldsymbol{d}_{0}}$.
Hence,

$\Big{(}H^{p-v}_{\mathcal{P}}(\boldsymbol{m}-\sum_{k\in
I}{\boldsymbol{d}_{k}})\otimes\bigwedge_{k\in
I}e_{k}\Big{)}\oplus\Big{(}H^{p-v}_{\mathcal{P}}(\boldsymbol{m}-\sum_{k\in
J}{\boldsymbol{d}_{k}})\otimes\bigwedge_{k\in J}e_{k}\Big{)}\cong\\\
H^{p-v}_{\mathcal{P}}\Big{(}\boldsymbol{m}-(\sum_{j=1}^{B}s_{j},\dots,\sum_{j=1}^{B}s_{j},s_{1},\dots,s_{B})-s_{0}\,{\boldsymbol{d}_{0}}\Big{)}\otimes\Big{(}\bigwedge_{k\in
I}e_{k}\oplus\bigwedge_{k\in J}e_{k}\Big{)}.$

By definition of $\mathcal{E}_{1},\dots,\mathcal{E}_{B}$ (def. 3.1), the set
$\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}$ is not empty if and only if $0\leq
s_{0}\leq 1$ and for all $i\in\\{1,\dots,B\\}$ it holds $0\leq
s_{i}\leq\mathcal{E}_{i}$. Hence,

$\\{I:I\subset\\{0,\dots,N\\},\\#I=p\\}=\bigcup\limits_{\begin{subarray}{c}0\leq
s_{0}\leq 1\\\ 0\leq s_{1}\leq\mathcal{E}_{1}\\\ \dots\\\ 0\leq
s_{B}\leq\mathcal{E}_{B}\\\
s_{0}+s_{1}+\dots+s_{B}=p\end{subarray}}\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}.$

Thus, (3.5) holds. The isomorphism in (16) follows from Prop. 2.7. ∎

In what follows, we identify the degree vectors that reduce the Weyman complex
to have just two elements and, in this way, they provide us Koszul-type
determinantal formulas for star multilinear systems (Def. 2.14). These degree
vectors are associated to tuples called _determinantal data_. The
determinantal data parameterize the different Koszul-type determinantal
formulas that we can obtain using the Weyman complex.

######  Definition 3.6.

Consider a partition of $\\{1,\dots,B\\}$ consisting of two sets $P$ and $D$
and a constant $c\in\mathbb{N}$. We say that the triplet $(P,D,c)$ is
_determinantal data_ in the following cases:

  * •

When $f_{0}$ corresponds to Center-Vertex or Edge case: if holds, $0\leq c\leq
A$.

  * •

When $f_{0}$ corresponds to Outer-Vertex case: if the following holds,

$\begin{cases}c=0&\text{if }\sum_{j\in P}d_{0,\boldsymbol{Y}_{j}}=0,\text{
or}\\\ c=A&\text{if }\sum_{j\in D}d_{0,\boldsymbol{Y}_{j}}=0.\end{cases}$

  * •

When $f_{0}$ corresponds to Triangle case: if the following holds,

$\left\\{\begin{array}[]{l}0\leq c\leq A,\\\ \sum_{j\in
P}d_{0,\boldsymbol{Y}_{j}}\leq 1,\text{ and}\\\ \sum_{j\in
D}d_{0,\boldsymbol{Y}_{j}}\leq 1.\end{array}\right.$

Equivalently, we say that the triplet $(P,D,c)$ is _determinantal data_ for
the multidegree ${\boldsymbol{d}_{0}}$ if the following conditions are
satisfied:

$\displaystyle\left\\{\begin{array}[]{c l}\sum_{j\in
P}d_{0,\boldsymbol{Y}_{j}}\leq 1\\\ \sum_{j\in D}d_{0,\boldsymbol{Y}_{j}}\leq
1\\\ 0\leq c\leq A&\text{ when }(\forall\,i\in[A])\,\text{ it holds
}d_{0,\boldsymbol{X}_{i}}=1,\\\ c=0&\text{ when }(\forall\,i\in[A])\,\text{ it
holds }d_{0,\boldsymbol{X}_{i}}=0\text{ and }\sum_{j\in
P}d_{0,\boldsymbol{Y}_{j}}=0,\\\ c=A&\text{ when }(\forall\,i\in[A])\,\text{
it holds }d_{0,\boldsymbol{X}_{i}}=0\text{ and }\sum_{j\in
D}d_{0,\boldsymbol{Y}_{j}}=0.\end{array}\right.$ (22)

######  Theorem 3.7.

 Consider a generic overdetermined system
$\boldsymbol{F}\\!=\\!(F_{0},\dots,F_{N})$  in
$\mathbb{Z}[\boldsymbol{u}][\text{\small$\boldsymbol{\bar{X},\bar{Y}}$}]$ of
multidegrees ${\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{N}}$ (Def. 2.3),
where $(F_{1},\dots,F_{n})$ is a square star multilinear system. Assume that
for every $j\in\\{1,\dots,B\\}$ it holds $\mathcal{E}_{j}\geq\beta_{j}$ (see
Rem. 3.3) and that the multidegree of $F_{0}$, ${\boldsymbol{d}_{0}}$, is a
solution of the system in (13). Then, for each determinantal data $(P,D,c)$
(Def. 3.6) and a permutation
$\sigma:\\{1,\dots,A\\}\rightarrow\\{1,\dots,A\\}$, the degree vector
$\boldsymbol{m}=(m_{\boldsymbol{X}_{1}},\dots,m_{\boldsymbol{X}_{A}},m_{\boldsymbol{Y}_{1}},\dots,m_{\boldsymbol{Y}_{B}})$,
defined by

$\displaystyle\left\\{\begin{array}[]{lcl}m_{\boldsymbol{X}_{i}}=\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{\sigma(i)-1}\alpha_{\sigma^{-1}(k)}+d_{0,\boldsymbol{X}_{i}}&&\text{for
}1\leq i\leq A\text{ and }\sigma(i)>c\\\ m_{\boldsymbol{X}_{i}}=\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{\sigma(i)-1}\alpha_{\sigma^{-1}(k)}-1&&\text{for
}1\leq i\leq A\text{ and }\sigma(i)\leq c\\\
m_{\boldsymbol{Y}_{j}}=\mathcal{E}_{j}-\beta_{j}+d_{0,{\boldsymbol{Y}_{j}}}&&\text{for
}j\in P\\\ m_{\boldsymbol{Y}_{j}}=-1&&\text{for }j\in D\end{array}\right.$

corresponds to the Koszul-type determinantal formula (Def. 2.14)

$K_{\bullet}(\boldsymbol{m}):0\rightarrow
K_{1,\omega+1}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\xrightarrow{\delta_{1}(\boldsymbol{m})}K_{0,\omega}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\rightarrow
0,$

where $\omega=\sum_{k=1}^{c}\alpha_{\sigma^{-1}(k)}+\sum_{j\in D}\beta_{j}$.

###### Proof.

To simplify the presentation of the proof, we assume with no loss of
generality that $\sigma$ is the identity map. We rewrite (7) using Lem. 3.5.
Hence, we obtain the following isomorphism,

$\displaystyle\small
H^{p-v}_{\mathcal{P}}\Big{(}\boldsymbol{m}-(\sum_{j=1}^{B}s_{j},\dots,\sum_{j=1}^{B}s_{j},s_{1},\dots,s_{B})-s_{0}\,{\boldsymbol{d}_{0}}\Big{)}\cong$
$\displaystyle\bigoplus_{\begin{subarray}{c}\sum_{i}r_{{\boldsymbol{X}_{i}}}+\sum_{j}r_{\boldsymbol{Y}_{j}}=p-v\end{subarray}}\left(\begin{array}[]{c
l}\displaystyle\bigotimes_{j\in
P}H^{r_{\boldsymbol{Y}_{j}}}_{\mathbb{P}^{\beta_{j}}}(\mathcal{E}_{j}-\beta_{j}+d_{0,{\boldsymbol{Y}_{j}}}-s_{j}-s_{0}\,d_{0,{\boldsymbol{Y}_{j}}})\;\otimes&\text{\footnotesize[Case
Y.1]}\\\ \displaystyle\bigotimes_{j\in
D}H^{r_{\boldsymbol{Y}_{j}}}_{\mathbb{P}^{\beta_{j}}}(-1-s_{j}-s_{0}\,d_{0,{\boldsymbol{Y}_{j}}})\;\otimes&\text{\footnotesize[Case
Y.2]}\\\
\displaystyle\bigotimes_{i=1}^{c}H^{r_{{\boldsymbol{X}_{i}}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}-1-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,\boldsymbol{X}_{i}}\Big{)}\;\otimes&\text{\footnotesize[Case
X.1]}\\\
\displaystyle\bigotimes_{i=c+1}^{A}\\!\\!H^{r_{{\boldsymbol{X}_{i}}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}+d_{0,\boldsymbol{X}_{i}}-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,{\boldsymbol{X}_{i}}}\\!\Big{)}&\text{\footnotesize[Case
X.2]}\end{array}\right)$ (27)

We will study the values for
$p,v,s_{0},\dots,s_{B},r_{{\boldsymbol{X}_{1}}},\dots,r_{{\boldsymbol{X}_{A}}},r_{\boldsymbol{Y}_{1}},\dots,r_{\boldsymbol{Y}_{B}}$
such that $K_{v,p}(\boldsymbol{m})$ (3.5) does not vanish. Clearly, if $0\leq
s_{0}\leq 1$ and $(\forall i\in\\{1,\dots,B\\})\;0\leq
s_{i}\leq\mathcal{E}_{i}$, then the module
$\bigoplus_{I\in\mathcal{I}_{s_{0},s_{1},\dots,s_{B}}}\bigwedge_{k\in I}e_{k}$
is not zero. Hence, assuming $0\leq s_{0}\leq 1$ and $(\forall
i\in\\{1,\dots,B\\})\;0\leq s_{i}\leq\mathcal{E}_{i}$, we study the vanishing
of the modules in (3.1). By Rem. 2.8, the modules in the right hand side of
(3.1) are not zero only when, for $1\leq i\leq A$,
$r_{\boldsymbol{X}_{i}}\in\\{0,\alpha_{i}\\}$ and, for $1\leq j\leq B$,
$r_{\boldsymbol{Y}_{j}}\in\\{0,\beta_{j}\\}$. We can use Rem. 2.9 to show that
if (3.1) does not vanish, then we have the following cases

$\displaystyle\hfill\begin{array}[]{| c | l | c| }\hline\cr\text{[Case Y.1]}&\text{For }j\in P&\begin{array}[]{c}r_{\boldsymbol{Y}_{j}}=0\text{ and }\mathcal{E}_{j}\\!-\\!\beta_{j}\\!+\\!d_{0,\boldsymbol{Y_{j}}}\geq s_{j}\\!+\\!s_{0}\,d_{0,\boldsymbol{Y_{j}}}\end{array}\\\\[10.03749pt] \hline\cr\text{[Case Y.2]}&\text{For }j\in D&\begin{array}[]{c}r_{\boldsymbol{Y}_{j}}=\beta_{j}\text{ and }s_{j}+s_{0}\,d_{0,\boldsymbol{Y_{j}}}\geq\beta_{j}\end{array}\\\\[10.03749pt] \hline\cr\text{[Case X.1]}&\text{For }1\leq i\leq c&\begin{array}[]{c}\displaystyle r_{\boldsymbol{X}_{i}}=\alpha_{i}\text{ and }\displaystyle\sum_{j=1}^{B}s_{j}+s_{0}\,d_{0,\boldsymbol{X}_{i}}\geq\sum_{j\in D}\beta_{j}+\sum_{k=1}^{i}\alpha_{\boldsymbol{X}_{k}}\end{array}\\\\[20.075pt] \hline\cr\text{[Case X.2]}&\text{For }c<i\leq A&\begin{array}[]{c}r_{\boldsymbol{X}_{i}}=0\text{ and }\displaystyle\sum_{j\in D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}\geq\sum_{j=1}^{B}s_{j}+(s_{0}-1)\,\,d_{0,\boldsymbol{X}_{i}}\end{array}\\\\[20.075pt] \hline\cr\end{array}$ (36)

From (36), we can deduce the possible values for $v$ such that
$K_{v,p}(\boldsymbol{m})$ does not vanish. From (3.5), it holds
$p=\sum_{j=1}^{B}s_{j}+s_{0}$. By Prop. 2.7,
$p-v=\sum_{i=1}^{A}r_{\boldsymbol{X}_{i}}+\sum_{j=1}^{B}r_{\boldsymbol{Y_{j}}}$.
Hence, we deduce that

$v=\sum_{j=1}^{B}s_{j}+s_{0}-\sum_{j\in D}\beta_{j}-\sum_{i=1}^{c}\alpha_{i}.$

  * •

First we provide a lower bound for $v$. Assume that $c>0$. By [Case X.1], if
$i=c$, then

$\sum_{j=1}^{B}s_{j}+s_{0}\,d_{0,\boldsymbol{X}_{c}}\geq\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{c}\alpha_{\boldsymbol{X}_{k}}.$

Hence, as $0\leq s_{0},d_{0,\boldsymbol{X}_{c+1}}\leq 1$, we conclude that
$v\geq 0$ as

$v=s_{0}+\sum_{j=1}^{B}s_{j}-\sum_{j\in
D}\beta_{j}-\sum_{k=1}^{c}\alpha_{k}\geq
s_{0}\,(1-d_{0,\boldsymbol{X}_{c}})\geq 0.$

Assume instead that $c=0$. Then $v=\sum_{j=1}^{B}s_{j}+s_{0}-\sum_{j\in
D}\beta_{j}$. By [Case Y.2], for each $j\in D$, $\beta_{j}\leq
s_{j}+s_{0}\,d_{0,\boldsymbol{Y_{j}}}$. Moreover, it holds, for each $j\in P$,
$0\leq s_{j}$. Adding the inequalities we deduce that,

$\displaystyle\sum_{j=1}^{B}s_{j}+s_{0}\,\sum_{j\in
D}d_{0,\boldsymbol{Y_{j}}}\geq\sum_{j\in D}\beta_{j}.$ (37)

By definition, $0\leq\sum_{j\in D}d_{0,\boldsymbol{Y_{j}}}\leq 1$. Hence, by
(37), $v\geq 0$ as, $v\geq s_{0}-s_{0}\sum_{j\in
D}d_{0,\boldsymbol{Y_{j}}}\geq 0.$

  * •

Finally we provide an upper bound for $v$. Assume that $c<A$. By [Case X.2],
if we consider $i=c+1$, then

$\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{c}\alpha_{k}\geq\sum_{j=1}^{B}s_{j}+(s_{0}-1)\,\,d_{0,\boldsymbol{X}_{c+1}}.$

Hence we conclude that $v\leq 1$, as $0\leq
s_{0},d_{0,\boldsymbol{X}_{c+1}}\leq 1$ and so,

$v=s_{0}+\sum_{j=1}^{B}s_{j}-\sum_{j\in
D}\beta_{j}-\sum_{k=1}^{c}\alpha_{k}\leq
s_{0}+(1-s_{0})\,d_{0,\boldsymbol{X}_{c+1}}\leq 1.$

Assume instead that $c=A$. Then $v=\sum_{j=1}^{B}s_{j}+s_{0}-\sum_{j\in
D}\beta_{j}-\sum_{i=1}^{A}\alpha_{i}$. By [Case Y.1], for $j\in P$,
$\mathcal{E}_{j}-\beta_{j}+d_{0,\boldsymbol{Y_{j}}}\geq
s_{j}+s_{0}\,d_{0,\boldsymbol{Y_{j}}}$. Moreover, it holds, for each $j\in D$,
$\mathcal{E}_{j}\geq s_{j}$. As the system $(F_{1},\dots,F_{N})$ is square, it
holds $\sum_{j=1}^{B}\mathcal{E}_{j}=\sum_{i=1}^{A}\alpha_{i}+\sum_{j\in
P}\beta_{j}+\sum_{j\in D}\beta_{j}$. Hence, adding the inequalities we obtain

$\displaystyle\sum_{i=1}^{A}\alpha_{i}+\sum_{j\in
D}\beta_{j}=\sum_{j=1}^{B}\mathcal{E}_{j}-\sum_{j\in
P}\beta_{j}\geq\sum_{j=1}^{B}s_{j}+(s_{0}-1)\sum_{j\in
P}d_{0,\boldsymbol{Y_{j}}}.$ (38)

By definition, $0\leq s_{0},\sum_{j\in P}d_{0,\boldsymbol{Y_{j}}}\leq 1$.
Hence, by (38), $v\leq 1$ as,

$v\leq s_{0}-(s_{0}-1)\,\sum_{j\in P}d_{0,\boldsymbol{Y_{j}}}\leq 1.$

We conclude that the possible values for $v$, $p$,
$r_{\boldsymbol{X}_{1}},\dots,r_{\boldsymbol{X}_{A}}$,
$r_{\boldsymbol{Y}_{1}},\dots,r_{\boldsymbol{Y}_{B}}$ such that (3.1) is not
zero are $v\in\\{0,1\\}$, the possible values for
$r_{\boldsymbol{X}_{1}},\dots,r_{\boldsymbol{X}_{A}}$,
$r_{\boldsymbol{Y}_{1}},\dots,r_{\boldsymbol{Y}_{B}}$ are the ones in (36) and
$p=\sum_{k=1}^{c}\alpha_{k}+\sum_{j\in D}\beta_{j}+v$. Let
$\omega=\sum_{k=1}^{c}\alpha_{k}+\sum_{j\in D}\beta_{j}$. Hence, our Weyman
complex looks like (9), where

$\delta_{1}(\boldsymbol{m}):K_{1,\omega+1}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}]\to
K_{0,\omega}(\boldsymbol{m})\otimes\mathbb{Z}[\boldsymbol{u}].$

In what follows, we prove each case in (36). Consider the modules related to
the variables $\boldsymbol{Y_{j}}$, for $j\in\\{1,\dots,B\\}$.

Case (Y.1) We consider the modules that involve the variables in the block
$\boldsymbol{Y_{j}}$, for $j\in P$. As $s_{j}\leq\mathcal{E}_{j}$ and
$s_{0},d_{0,\boldsymbol{Y_{j}}}\leq 1$, it holds
$\mathcal{E}_{j}-\beta_{j}+d_{0,\boldsymbol{Y_{j}}}-s_{j}-s_{0}\,d_{0,\boldsymbol{Y_{j}}}>-\beta_{j}-1$.
Hence, by Rem. 2.9,

$\displaystyle
H^{r_{\boldsymbol{Y}_{j}}}_{\mathbb{P}^{\beta_{j}}}(\mathcal{E}_{j}-\beta_{j}+d_{0,\boldsymbol{Y_{j}}}-s_{j}-s_{0}\,d_{0,\boldsymbol{Y_{j}}})\neq
0\iff$ (39) $\displaystyle r_{\boldsymbol{Y}_{j}}=0\text{ and
}\mathcal{E}_{j}-\beta_{j}+d_{0,\boldsymbol{Y_{j}}}\geq
s_{j}+s_{0}\,d_{0,\boldsymbol{Y_{j}}}.$

Case (Y.2) We consider the modules that involve the variables in the block
$\boldsymbol{Y_{j}}$, for $j\in D$. As
$s_{j},s_{0},d_{0,\boldsymbol{Y_{j}}}\geq 0$, then
$-1-s_{j}-s_{0}\,d_{0,\boldsymbol{Y_{j}}}<0$. Hence, by Rem. 2.9,

$\displaystyle
H^{r_{\boldsymbol{Y}_{j}}}_{\mathbb{P}^{\beta_{j}}}(-1-s_{j}-s_{0}\,d_{0,\boldsymbol{Y_{j}}})\neq
0\iff r_{\boldsymbol{Y}_{j}}=\beta_{j}\quad\text{ and }\quad
s_{j}+s_{0}\,d_{0,\boldsymbol{Y_{j}}}\geq\beta_{j}.$ (40)

Now we consider the cohomologies related to the blocks of variables
$\boldsymbol{X_{i}}$, for $i\in\\{1,\dots,A\\}$. We assume that the
cohomologies related to the blocks of variables $\boldsymbol{Y_{j}}$ do not
vanish.

Case (X.1) We consider the module related to the blocks
$\boldsymbol{X}_{1}\dots,\boldsymbol{X}_{c}$. We only need to consider this
case if $c>0$, so we assume $c>0$. We prove that for each $1\leq i\leq c$, if
the cohomologies related to the variables in the blocks $\boldsymbol{Y}_{j}$,
for $1\leq j\leq B$, and the ones related to
$\boldsymbol{X}_{1}\dots,\boldsymbol{X}_{i-1}$, do not vanish, then

$\displaystyle\begin{gathered}H^{r_{\boldsymbol{X}_{k}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}m_{\boldsymbol{X}_{i}}-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,\boldsymbol{X}_{i}}\Big{)}\neq
0\iff\\\ \quad r_{\boldsymbol{X}_{i}}=\alpha_{i}\text{ and
}\sum_{j=1}^{B}s_{j}+s_{0}\,d_{0,\boldsymbol{X}_{i}}\geq\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i}\alpha_{\boldsymbol{X}_{k}}.\end{gathered}$ (43)

We proceed by induction on $1\leq i\leq c$.

  * •

Consider $i=1$ and the cohomology related to the block $\boldsymbol{X}_{1}$,

$H^{r_{\boldsymbol{X}_{1}}}_{\mathbb{P}^{\alpha_{1}}}\Big{(}\sum_{j\in
D}\beta_{j}-1-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,\boldsymbol{X}_{1}}\Big{)}.$

As we assumed that $c>0$ and the triplet $(P,D,c)$ is determinantal data (def.
3.6), by definition either $d_{0,\boldsymbol{X}_{1}}=1$ or both
$d_{0,\boldsymbol{X}_{1}}=0$ and $\sum_{j\in D}d_{0,\boldsymbol{Y_{j}}}=0$.
Also, it holds $0\leq s_{0},\sum_{j\in D}d_{0,\boldsymbol{Y_{j}}}\leq 1$.
Hence, from (37), we conclude that,

$\sum_{j\in
D}\beta_{j}-1-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,\boldsymbol{X}_{k}}\leq
s_{0}\,\sum_{j\in
D}d_{0,\boldsymbol{Y_{j}}}-1-s_{0}\,d_{0,\boldsymbol{X}_{k}}<0$

Therefore, by Rem. 2.9,

$\displaystyle\begin{gathered}H^{r_{\boldsymbol{X}_{1}}}_{\mathbb{P}^{\alpha_{1}}}\Big{(}\sum_{j\in
D}\beta_{j}-1-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,\boldsymbol{X}_{1}}\Big{)}\neq
0\iff\\\ r_{\boldsymbol{X}_{1}}=\alpha_{1}\quad\text{ and
}\quad\sum_{j=1}^{B}s_{j}+s_{0}\,d_{0,\boldsymbol{X}_{1}}\geq\sum_{j\in
D}\beta_{j}+\alpha_{\boldsymbol{X}_{1}}\end{gathered}$

  * •

We proceed by induction, assuming that (43) holds for $i-1$, we prove the
property for $i$. We consider the cohomology

$H^{r_{\boldsymbol{X}_{i}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}\sum_{j\in
D}\beta_{j}-1+\sum_{k=1}^{i-1}\alpha_{k}-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,\boldsymbol{X}_{i}}\Big{)}.$

By definition (see (13)),
$d_{0,\boldsymbol{X}_{i-1}}=d_{0,\boldsymbol{X}_{i}}$, and by inductive
hypothesis, if the previous modules do not vanish, then

$\sum_{j=1}^{B}s_{j}+s_{0}\,d_{0,\boldsymbol{X}_{i}}=\sum_{j=1}^{B}s_{j}+s_{0}\,d_{0,\boldsymbol{X}_{i-1}}\geq\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{\boldsymbol{X}_{k}}.$

Hence, by Rem. 2.9,

$\displaystyle\begin{gathered}H^{r_{\boldsymbol{X}_{i}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}\sum_{j\in
D}\beta_{j}-1+\sum_{k=1}^{i-1}\alpha_{k}-\sum_{j=1}^{B}s_{j}-s_{0}\,d_{0,\boldsymbol{X}_{i}}\Big{)}\neq
0\iff\\\
r_{\boldsymbol{X}_{i}}=\alpha_{i}\quad\text{and}\quad\sum_{j=1}^{B}s_{j}+s_{0}\,d_{0,\boldsymbol{X}_{i}}\geq\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}+\alpha_{i}.\end{gathered}$

Case (X.2) We consider the module related to the blocks
$\boldsymbol{X}_{c+1}\dots,\boldsymbol{X}_{A}$. We only need to consider this
case if $c<A$, so we assume $c<A$. We prove that for each $c<i\leq A$, if the
cohomologies related to the variables in the blocks $\boldsymbol{Y}_{j}$, for
$1\leq j\leq B$, and the ones related to
$\boldsymbol{X}_{i+1},\boldsymbol{X}_{i+2},\dots,\boldsymbol{X}_{A}$, do not
vanish, then

$\displaystyle\begin{gathered}H^{r_{\boldsymbol{X}_{i}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}+(1-s_{0})\,d_{0,\boldsymbol{X}_{i}}-\sum_{j=1}^{B}s_{j}\Big{)}\neq
0\iff\\\ r_{\boldsymbol{X}_{i}}=0\quad\text{and}\quad\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}\geq\sum_{j=1}^{B}s_{j}+(s_{0}-1)\,\,d_{0,\boldsymbol{X}_{i}}.\end{gathered}$
(46)

We proceed by induction.

  * •

Consider $i=A$ and the cohomology related to the block $\boldsymbol{X}_{A}$,

$H^{r_{\boldsymbol{X}_{A}}}_{\mathbb{P}^{\alpha_{A}}}\Big{(}\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{A-1}\alpha_{k}+(1-s_{0})\,d_{0,\boldsymbol{X}_{A}}-\sum_{j=1}^{B}s_{j}\Big{)}.$

As we assumed that $c<A$ and the triplet $(P,D,c)$ is determinantal data (Def.
3.6), by definition, either $d_{0,\boldsymbol{X}_{A}}=1$ or both
$d_{0,\boldsymbol{X}_{A}}=0$ and $\sum_{j\in P}d_{0,\boldsymbol{Y_{j}}}=0$.
Also it holds $0\leq s_{0},\sum_{j\in P}d_{0,\boldsymbol{Y_{j}}}\leq 1$.
Hence, from (38), we conclude that

$\sum_{j\in
D}\beta_{j}+\sum_{i=1}^{A-1}\alpha_{i}+(1-s_{0})\,d_{0,\boldsymbol{X}_{A}}-\sum_{j=1}^{B}s_{j}\geq-\alpha_{A}$

Therefore, by Rem. 2.9,

$\displaystyle
H^{r_{\boldsymbol{X}_{A}}}_{\mathbb{P}^{\alpha_{A}}}\Big{(}\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{A-1}\alpha_{k}+(1-s_{0})\,d_{0,\boldsymbol{X}_{A}}-\sum_{j=1}^{B}s_{j}\Big{)}\neq
0\iff$ $\displaystyle r_{\boldsymbol{X}_{A}}=0\text{ and }\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{A-1}\alpha_{k}\geq\sum_{j=1}^{B}s_{j}+(s_{0}-1)\,\,d_{0,\boldsymbol{X}_{A}}.$
(47)

  * •

We proceed by induction, assuming that (46) holds for $i+1\leq A$, we prove
the property for $i>c$. We consider the cohomology

$H^{r_{\boldsymbol{X}_{i}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}+(1-s_{0})\,d_{0,\boldsymbol{X}_{i}}-\sum_{j=1}^{B}s_{j}\Big{)}.$

By definition (see (13)),
$d_{0,\boldsymbol{X}_{i+1}}=d_{0,\boldsymbol{X}_{i}}$. So, if the previous
cohomologies do not vanish, by induction hypothesis,

$\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i}\alpha_{k}\geq(s_{0}-1)\,\,d_{0,\boldsymbol{X}_{{i+1}}}+\sum_{j=1}^{B}s_{j}=(s_{0}-1)\,\,d_{0,\boldsymbol{X}_{{i}}}+\sum_{j=1}^{B}s_{j}$

Equivalently,

$\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}+(1-s_{0})\,\,d_{0,\boldsymbol{X}_{{i}}}-\sum_{j=1}^{B}s_{j}\geq-\alpha_{i}.$

Hence, by Rem. 2.9,

$\displaystyle\begin{gathered}H^{r_{\boldsymbol{X}_{i}}}_{\mathbb{P}^{\alpha_{i}}}\Big{(}\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}+(1-s_{0})\,d_{0,\boldsymbol{X}_{i}}-\sum_{j=1}^{B}s_{j}\Big{)}\neq
0\iff\\\ r_{\boldsymbol{X}_{i}}=0\quad\text{and}\quad\sum_{j\in
D}\beta_{j}+\sum_{k=1}^{i-1}\alpha_{k}\geq\sum_{j=1}^{B}s_{j}+(s_{0}-1)\,\,d_{0,\boldsymbol{X}_{i}}.\end{gathered}$

∎

The previous theorem gives us Sylvester-like determinantal formulas in some
cases.

######  Corollary 3.8 (Sylvester-type formulas).

Consider $\boldsymbol{d}_{0}$ corresponding to the Center-Vertex or Edge case.
Let $\sigma:\\{1,\dots,A\\}\rightarrow\\{1,\dots,A\\}$ be any permutation and
consider the determinantal data $(\\{1,\dots,B\\},\emptyset,0)$. Then, by
Prop. 2.17, the overdetermined systems from Thm. 3.7 have a _Sylvester-like
formula_ coming from the degree vector $\boldsymbol{m}$ related to the
determinantal data $(\\{1,\dots,B\\},\emptyset,0)$ and the permutation
$\sigma$.

For each determinantal formula given by Thm. 3.7, we get another one from its
dual.

######  Remark 3.9.

Consider a degree vector $\boldsymbol{m}$ related to the determinantal data
$(P,D,c)$ and the permutation $\sigma$. Then, the triplet $(D,P,A-c)$ is also
determinantal data and the map $i\mapsto(A+1-\sigma(i))$ is a permutation of
$\\{1,\dots,A\\}$. Let $\boldsymbol{\bar{m}}$ be the degree vector associated
to $(D,P,A-c)$ and $i\mapsto(A+1-\sigma(i))$, then, by Prop. 2.10,
$K_{\bullet}(\boldsymbol{m})$ is isomorphic to the dual complex of
$K_{\bullet}(\boldsymbol{\bar{m}})$.

###  3.2 Size of determinantal formulas

Following general approaches for resultants as in [14, Ch. 3] or specific
ideas for Koszul-type formulas as in [5], we can use the matrices associated
to the determinantal formulas from Thm. 3.7 to solve the square systems
$(f_{1},\dots,f_{N})$. To express the complexity of these approaches in terms
of the size of the output, that is, the expected number of solutions, we study
the size of the determinantal formulas of Thm. 3.7 and we compare them with
the number of solutions of the system.

The multihomogeneous Bézout bound (Prop. 2.2) implies the following lemma.

######  Lemma 3.10.

 The expected number of solutions,  $\Upsilon$,  of a square star
multilinear system is

$\Upsilon:=\frac{(\sum_{i=1}^{A}\alpha_{i})!}{\prod_{i=1}^{A}\alpha_{i}!}\cdot\prod_{j=1}^{B}{\mathcal{E}_{j}\choose\beta_{j}}.$

######  Lemma 3.11.

The degree of the resultant and the sizes of the matrices corresponding to the
determinantal formulas of Thm. 3.7, that is, the rank of the modules
$K_{0}(\boldsymbol{m})$ and $K_{1}(\boldsymbol{m})$, are (See the beginning of
Sec. 3.1 for the definition of the four cases and the notation in the bounds)
as follows:

  * •

Center-Vertex case: The rank of the modules is
$\displaystyle\Upsilon\cdot(1+\sum\nolimits_{i=1}^{A}\alpha_{i})$.

  * •

Outer-Vertex case: The rank of the modules is
$\displaystyle\Upsilon\cdot\frac{\mathcal{E}_{j}+\beta_{j}\,(\sum\nolimits_{i=1}^{A}\alpha_{i})+1}{\mathcal{E}_{j}-\beta_{j}+1}$.

  * •

Edge case: The rank of the modules is
$\displaystyle\Upsilon\cdot\frac{(1+\sum\nolimits_{i=1}^{A}\alpha_{i})(\mathcal{E}_{j}+1)}{\mathcal{E}_{j}-\beta_{j}+1}.$

  * •

Triangle case: The rank is
$\displaystyle\Upsilon\cdot(1+\sum\nolimits_{i=1}^{A}\alpha_{i})\,(1+\frac{\beta_{j_{1}}}{\mathcal{E}_{j_{1}}-\beta_{j_{1}}+1}+\frac{\beta_{j_{2}}}{\mathcal{E}_{j_{2}}-\beta_{j_{2}}+1}).$

The proof of this lemma can be found in Sec. A.1 and follows from a direct
computation, see Prop. 2.4.

###  3.3 Example

We follow the notation from the beginning of Sec. 3. Consider four blocks of
variables
$\boldsymbol{X_{1}},\boldsymbol{X_{2}},\boldsymbol{Y_{1}},\boldsymbol{Y_{2}}$
that we partition to two sets: $\\{\boldsymbol{X_{1}},\boldsymbol{X_{2}}\\}$,
of cardinality $A=2$, and $\\{\boldsymbol{Y_{1}},\boldsymbol{Y_{2}}\\}$, of
cardinality $B=2$. The number of variables in the blocks of the first set are
$\alpha=(1,1)$ and in the second $\beta=(1,1)$. That is, we consider

$\boldsymbol{X_{1}}:=\\{X_{1,0},X_{1,1}\\},\boldsymbol{X_{2}}:=\\{X_{2,0},X_{2,1}\\},\quad\boldsymbol{Y_{1}}:=\\{Y_{1,0},Y_{1,1}\\},\boldsymbol{Y_{2}}:=\\{Y_{2,0},Y_{2,1}\\}.$

Let $(f_{1},\dots,f_{4})$ be a square star multilinear system corresponding to
the following graph,

${\mathbb{K}[{\boldsymbol{X}_{1}},{\boldsymbol{X}_{2}}]}$${\mathbb{K}[\boldsymbol{Y}_{1}]}$${\mathbb{K}[\boldsymbol{Y}_{2}]}$$\mathcal{E}_{1}=2$$\mathcal{E}_{2}=2$

By Lem. 3.10, the expected number of solutions of the system is $8$. We
introduce a polynomial $f_{0}$ and we consider the multiprojective resultant
of $\boldsymbol{f}:=(f_{0},f_{1},\dots,f_{N})$. By Lem. 3.11, the degree of
the resultant, depending on the choice of $f_{0}$, is as follows:

  * •

In the Center-Vertex case, that is
$f_{0}\in\mathbb{K}[\boldsymbol{X}_{1},\boldsymbol{X}_{2}]_{\boldsymbol{1}}$,
the degree of the resultant is $24$.

  * •

In the Outer-Vertex case, that is
$f_{0}\in\mathbb{K}[\boldsymbol{Y}_{j}]_{\boldsymbol{1}}$ for $j\in[2]$, its
degree is $20$.

  * •

In the Edge case, that is
$f_{0}\in\mathbb{K}[\boldsymbol{X}_{1},\boldsymbol{X}_{2},\boldsymbol{Y}_{j}]_{\boldsymbol{1}}$
for $j\in[2]$, its degree is $36$.

  * •

In the Triangle case, that is
$f_{0}\in\mathbb{K}[\boldsymbol{X}_{1},\boldsymbol{X}_{2},\boldsymbol{Y}_{1},\boldsymbol{Y}_{2}]_{\boldsymbol{1}}$,
its degree is $48$.

We consider the Edge case, where
$f_{0}\in\mathbb{K}[\boldsymbol{X}_{1},\boldsymbol{X}_{2}]_{\boldsymbol{1}}$,
and the overdetermined system
$\boldsymbol{f}=(f_{0},f_{1},f_{2},f_{3},f_{4},f_{5})$, where,

$\displaystyle\begin{array}[]{l
l}f_{0}\\!\\!:=\\!\\!\\!\\!\\!\\!&\left(a_{{1}}\,x_{2;{0}}\\!+\\!a_{{2}}\,x_{2;{1}}\right)x_{1;{0}}\\!+\\!\left(a_{{3}}\,x_{2;{0}}\\!+\\!a_{{4}}\,x_{2;{1}}\right)x_{1;{1}}\\\\[7.0pt]
f_{1}\\!\\!:=\\!\\!\\!\\!\\!\\!&\left(\left(b_{{1}}\,y_{1;{0}}\\!+\\!b_{{2}}\,y_{1;{1}}\right)x_{2;{0}}\\!+\\!\left(b_{{3}}\,y_{1;{0}}\\!+\\!b_{{4}}\,y_{1;{1}}\right)x_{2;{1}}\right)x_{1;{0}}\\!+\\!\left(\left(b_{{5}}\,y_{1;{0}}\\!+\\!b_{{6}}\,y_{1;{1}}\right)x_{2;{0}}\\!+\\!\left(b_{{7}}\,y_{1;{0}}\\!+\\!b_{{8}}\,y_{1;{1}}\right)x_{2;{1}}\right)x_{1;{1}}\\\\[7.0pt]
f_{2}\\!\\!:=\\!\\!\\!\\!\\!\\!&\left(\left(c_{{1}}\,y_{1;{0}}\\!+\\!c_{{2}}\,y_{1;{1}}\right)x_{2;{0}}\\!+\\!\left(c_{{3}}\,y_{1;{0}}\\!+\\!c_{{4}}\,y_{1;{1}}\right)x_{2;{1}}\right)x_{1;{0}}\\!+\\!\left(\left(c_{{5}}\,y_{1;{0}}\\!+\\!c_{{6}}\,y_{1;{1}}\right)x_{2;{0}}\\!+\\!\left(c_{{7}}\,y_{1;{0}}\\!+\\!c_{{8}}\,y_{1;{1}}\right)x_{2;{1}}\right)x_{1;{1}}\\\\[7.0pt]
f_{3}\\!\\!:=\\!\\!\\!\\!\\!\\!&\left(\left(d_{{1}}\,y_{2;{0}}\\!+\\!d_{{2}}\,y_{2;{1}}\right)x_{2;{0}}\\!+\\!\left(d_{{3}}\,y_{2;{0}}\\!+\\!d_{{4}}\,y_{2;{1}}\right)x_{2;{1}}\right)x_{1;{0}}\\!+\\!\left(\left(d_{{5}}\,y_{2;{0}}\\!+\\!d_{{6}}\,y_{2;{1}}\right)x_{2;{0}}\\!+\\!\left(d_{{7}}\,y_{2;{0}}\\!+\\!d_{{8}}\,y_{2;{1}}\right)x_{2;{1}}\right)x_{1;{1}}\\\\[7.0pt]
f_{4}\\!\\!:=\\!\\!\\!\\!\\!\\!&\left(\left(e_{{1}}\,y_{2;{0}}\\!+\\!e_{{2}}\,y_{2;{1}}\right)x_{2;{0}}\\!+\\!\left(e_{{3}}\,y_{2;{0}}\\!+\\!e_{{4}}\,y_{2;{1}}\right)x_{2;{1}}\right)x_{1;{0}}\\!+\\!\left(\left(e_{{5}}\,y_{2;{0}}\\!+\\!e_{{6}}\,y_{2;{1}}\right)x_{2;{0}}\\!+\\!\left(e_{{7}}\,y_{2;{0}}\\!+\\!e_{{8}}\,y_{2;{1}}\right)x_{2;{1}}\right)x_{1;{1}}\end{array}$

We consider the _determinantal data_ $(\\{1\\},\\{2\\},1)$ and the identity
map $i\mapsto i$. Then, the degree vector of Thm. 3.7 is
$\boldsymbol{m}=(0,3,1,-1)$ and the vector spaces of the Weyman complex
$K(\boldsymbol{m},\boldsymbol{f})$ become

$\displaystyle K_{1}(\boldsymbol{m},\boldsymbol{f})=$
$\displaystyle\quad\,\,S^{*}_{\boldsymbol{X}_{1}}(-1)\otimes
S_{\boldsymbol{X}_{2}}(0)\otimes S_{\boldsymbol{Y}_{1}}(0)\otimes
S^{*}_{\boldsymbol{Y}_{2}}(0)\otimes\left\\{\begin{array}[]{l}(e_{0}\wedge
e_{1}\wedge e_{3})\oplus(e_{0}\wedge e_{1}\wedge e_{4})\;\oplus\\\
(e_{0}\wedge e_{2}\wedge e_{3})\oplus(e_{0}\wedge e_{2}\wedge
e_{4})\end{array}\right\\}$
$\displaystyle\oplus\;S^{*}_{\boldsymbol{X}_{1}}(-1)\otimes
S_{\boldsymbol{X}_{2}}(0)\otimes S_{\boldsymbol{Y}_{1}}(0)\otimes
S^{*}_{\boldsymbol{Y}_{2}}(-1)\otimes\left\\{\begin{array}[]{l}(e_{1}\wedge
e_{3}\wedge e_{4})\oplus(e_{2}\wedge e_{3}\wedge e_{4})\end{array}\right\\}$
$\displaystyle\oplus\;S^{*}_{\boldsymbol{X}_{1}}(-1)\otimes
S_{\boldsymbol{X}_{2}}(0)\otimes S_{\boldsymbol{Y}_{1}}(1)\otimes
S^{*}_{\boldsymbol{Y}_{2}}(-1)\otimes\left\\{\begin{array}[]{l}(e_{0}\wedge
e_{3}\wedge e_{4})\end{array}\right\\},$ $\displaystyle
K_{0}(\boldsymbol{m},\boldsymbol{f})=$
$\displaystyle\quad\,\,S^{*}_{\boldsymbol{X}_{1}}(0)\otimes
S_{\boldsymbol{X}_{2}}(1)\otimes S_{\boldsymbol{Y}_{1}}(0)\otimes
S^{*}_{\boldsymbol{Y}_{2}}(0)\otimes\left\\{\begin{array}[]{l}(e_{1}\wedge
e_{3})\oplus(e_{1}\wedge e_{4})\;\oplus\\\ (e_{2}\wedge
e_{3})\oplus(e_{2}\wedge e_{4})\end{array}\right\\}$
$\displaystyle\oplus\;S^{*}_{\boldsymbol{X}_{1}}(0)\otimes
S_{\boldsymbol{X}_{2}}(1)\otimes S_{\boldsymbol{Y}_{1}}(1)\otimes
S^{*}_{\boldsymbol{Y}_{2}}(0)\otimes\left\\{\begin{array}[]{l}(e_{0}\wedge
e_{3})\oplus(e_{0}\wedge e_{4})\end{array}\right\\}$
$\displaystyle\oplus\;S^{*}_{\boldsymbol{X}_{1}}(0)\otimes
S_{\boldsymbol{X}_{2}}(1)\otimes S_{\boldsymbol{Y}_{1}}(1)\otimes
S^{*}_{\boldsymbol{Y}_{2}}(-1)\otimes\left\\{\begin{array}[]{l}(e_{3}\wedge
e_{4})\end{array}\right\\}.$

The Koszul determinantal matrix representing the map
$\delta_{1}(\boldsymbol{m},\boldsymbol{f})$ between the modules with respect
to the monomial basis is

$\displaystyle\left[\begin{array}[]{cccccccccccccccccccccccc}a_{{1}}&a_{{2}}&&&&&&&-b_{{1}}&-b_{{2}}&-b_{{3}}&-b_{{4}}&&&&&&&&&&&&\\\
a_{{3}}&a_{{4}}&&&&&&&-b_{{5}}&-b_{{6}}&-b_{{7}}&-b_{{8}}&&&&&&&&&&&&\\\
&&a_{{1}}&a_{{2}}&&&&&&&&&-b_{{1}}&-b_{{2}}&-b_{{3}}&-b_{{4}}&&&&&&&&\\\
&&a_{{3}}&a_{{4}}&&&&&&&&&-b_{{5}}&-b_{{6}}&-b_{{7}}&-b_{{8}}&&&&&&&&\\\
&&&&a_{{1}}&a_{{2}}&&&-c_{{1}}&-c_{{2}}&-c_{{3}}&-c_{{4}}&&&&&&&&&&&&\\\
&&&&a_{{3}}&a_{{4}}&&&-c_{{5}}&-c_{{6}}&-c_{{7}}&-c_{{8}}&&&&&&&&&&&&\\\
&&&&&&a_{{1}}&a_{{2}}&&&&&-c_{{1}}&-c_{{2}}&-c_{{3}}&-c_{{4}}&&&&&&&&\\\
&&&&&&a_{{3}}&a_{{4}}&&&&&-c_{{5}}&-c_{{6}}&-c_{{7}}&-c_{{8}}&&&&&&&&\\\
e_{{1}}&e_{{3}}&-d_{{1}}&-d_{{3}}&&&&&&&&&&&&&b_{{1}}&b_{{2}}&b_{{3}}&b_{{4}}&&&&\\\
e_{{5}}&e_{{7}}&-d_{{5}}&-d_{{7}}&&&&&&&&&&&&&b_{{5}}&b_{{6}}&b_{{7}}&b_{{8}}&&&&\\\
e_{{2}}&e_{{4}}&-d_{{2}}&-d_{{4}}&&&&&&&&&&&&&&&&&b_{{1}}&b_{{2}}&b_{{3}}&b_{{4}}\\\
e_{{6}}&e_{{8}}&-d_{{6}}&-d_{{8}}&&&&&&&&&&&&&&&&&b_{{5}}&b_{{6}}&b_{{7}}&b_{{8}}\\\
&&&&e_{{1}}&e_{{3}}&-d_{{1}}&-d_{{3}}&&&&&&&&&c_{{1}}&c_{{2}}&c_{{3}}&c_{{4}}&&&&\\\
&&&&e_{{5}}&e_{{7}}&-d_{{5}}&-d_{{7}}&&&&&&&&&c_{{5}}&c_{{6}}&c_{{7}}&c_{{8}}&&&&\\\
&&&&e_{{2}}&e_{{4}}&-d_{{2}}&-d_{{4}}&&&&&&&&&&&&&c_{{1}}&c_{{2}}&c_{{3}}&c_{{4}}\\\
&&&&e_{{6}}&e_{{8}}&-d_{{6}}&-d_{{8}}&&&&&&&&&&&&&c_{{5}}&c_{{6}}&c_{{7}}&c_{{8}}\\\
&&&&&&&&e_{{1}}&&e_{{3}}&&-d_{{1}}&&-d_{{3}}&&a_{{1}}&&a_{{2}}&&&&&\\\
&&&&&&&&e_{{5}}&&e_{{7}}&&-d_{{5}}&&-d_{{7}}&&a_{{3}}&&a_{{4}}&&&&&\\\
&&&&&&&&e_{{2}}&&e_{{4}}&&-d_{{2}}&&-d_{{4}}&&&&&&a_{{1}}&&a_{{2}}&\\\
&&&&&&&&e_{{6}}&&e_{{8}}&&-d_{{6}}&&-d_{{8}}&&&&&&a_{{3}}&&a_{{4}}&\\\
&&&&&&&&&e_{{1}}&&e_{{3}}&&-d_{{1}}&&-d_{{3}}&&a_{{1}}&&a_{{2}}&&&&\\\
&&&&&&&&&e_{{5}}&&e_{{7}}&&-d_{{5}}&&-d_{{7}}&&a_{{3}}&&a_{{4}}&&&&\\\
&&&&&&&&&e_{{2}}&&e_{{4}}&&-d_{{2}}&&-d_{{4}}&&&&&&a_{{1}}&&a_{{2}}\\\
&&&&&&&&&e_{{6}}&&e_{{8}}&&-d_{{6}}&&-d_{{8}}&&&&&&a_{{3}}&&a_{{4}}\end{array}\right].$

##  4 Determinantal formulas for bipartite bilinear system

In this section, we define the _bipartite bilinear systems_ and we construct
determinantal formulas for two different kinds of overdetermined
multihomogeneous systems related to them. These formulas are Koszul-type
determinantal formulas. This section follow the same notation as Sec. 3.

######  Definition 4.1 (Bipartite bilinear system).

A square multihomogeneous system $\boldsymbol{f}=(f_{1},\dots,f_{N})$ in
$\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$ with multidegrees
${\boldsymbol{d}_{1}},\dots,{\boldsymbol{d}_{N}}\in\mathbb{Z}^{q}$ is a
_bipartite bilinear system_ if for every $k\in[N]$, there are $i_{k}\in[A]$
and $j_{k}\in[B]$ such that
$f_{k}\in\mathbb{K}[{\boldsymbol{X}_{i_{k}}},{\boldsymbol{Y}_{j_{k}}}]_{\boldsymbol{1}}.$
For each $i\in[A]$ and $j\in[B]$, let $\mathcal{E}_{i,j}$ be the number of
polynomials of $\boldsymbol{f}$ in
$\mathbb{K}[{\boldsymbol{X}_{i}},{\boldsymbol{Y}_{\\!j}}]_{\boldsymbol{1}}.$

We use the term _bipartite_ because we can represent such systems using a
bipartite graph. The vertices of the graph are the algebras
${\mathbb{K}[\boldsymbol{X}_{1}]},\dots,{\mathbb{K}[\boldsymbol{X}_{A}]}$,
${\mathbb{K}[\boldsymbol{Y}_{1}]},\dots,{\mathbb{K}[\boldsymbol{Y}_{B}]}$. For
each ${\boldsymbol{d}_{k}}$ there is an edge between the vertices
${\mathbb{K}[\boldsymbol{X}_{i}]}$ and ${\mathbb{K}[\boldsymbol{Y}_{j}]}$
whenever $d_{k,X_{i}}=d_{k,Y_{j}}=1$. That is, when it holds
$f_{k}\in\mathbb{K}[\boldsymbol{X}_{i},\boldsymbol{Y}_{j}]_{\boldsymbol{1}}$.
The graph is bipartite because we can partition the vertices to two sets,
$\\{{\mathbb{K}[\boldsymbol{X}_{1}]},\dots,{\mathbb{K}[\boldsymbol{X}_{A}]}\\}$
and
$\\{{\mathbb{K}[\boldsymbol{Y}_{1}]},\dots,{\mathbb{K}[\boldsymbol{Y}_{B}]}\\}$
such that there is no edge between vertices belonging to the same set.

######  Example 4.2.

Let
${\boldsymbol{X}_{1}},{\boldsymbol{X}_{2}},{\boldsymbol{Y}_{1}},{\boldsymbol{Y}_{2}},{\boldsymbol{Y}_{3}}$
be five blocks of variables. Consider the multihomogeneous system
$(f_{1},f_{2},f_{3},f_{4})\subset\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$
with multidegrees

$\displaystyle\begin{array}[]{r c c | c c c c c}(\\!\\!\\!\\!&d_{i,X_{1}},&d_{i,X_{2}},&d_{i,Y_{1}},&d_{i,Y_{2}},&d_{i,Y_{3}}&\\!\\!\\!\\!)\\\ {\boldsymbol{d}_{1}}=(\\!\\!\\!\\!&1,&0,&1,&0,&0&\\!\\!\\!\\!)\\\ {\boldsymbol{d}_{2}}=(\\!\\!\\!\\!&1,&0,&0,&1,&0&\\!\\!\\!\\!)\\\ {\boldsymbol{d}_{3}}=(\\!\\!\\!\\!&0,&1,&0,&1,&0&\\!\\!\\!\\!)\\\ {\boldsymbol{d}_{4}}=(\\!\\!\\!\\!&0,&1,&0,&0,&1&\\!\\!\\!\\!)\end{array}$ ${\mathbb{K}[\boldsymbol{X}_{1}]}$${\mathbb{K}[\boldsymbol{X}_{2}]}$${\mathbb{K}[\boldsymbol{Y}_{1}]}$${\mathbb{K}[\boldsymbol{Y}_{2}]}$${\mathbb{K}[\boldsymbol{Y}_{3}]}$$\mathcal{E}_{1,1}=1$$\mathcal{E}_{1,2}=1$$\mathcal{E}_{2,2}=1$$\mathcal{E}_{2,3}=1$$\mathcal{E}_{1,3}=0$$\mathcal{E}_{2,1}=0$

This system is a bipartite bilinear system where $\mathcal{E}_{1,1}=1$,
$\mathcal{E}_{1,2}=1$, $\mathcal{E}_{1,3}=0$, $\mathcal{E}_{2,1}=0$,
$\mathcal{E}_{2,2}=1$ and $\mathcal{E}_{2,3}=1$. The corresponding bipartite
graph is the one above.

######  Remark 4.3.

For each square bipartite bilinear system, it holds
$N=\sum_{i=1}^{A}\sum_{j=1}^{B}\mathcal{E}_{i,j}$. Moreover, if the system has
a nonzero finite number of solutions, then for $i\in\\{1,\dots,A\\}$, it holds
$\sum_{j=1}^{B}\mathcal{E}_{i,j}\geq\alpha_{i}$ and for each
$j\in\\{1,\dots,B\\}$ it holds $\sum_{i=1}^{A}\mathcal{E}_{i,j}\geq\beta_{j}$,
see Prop. 2.2.

As we did in Sec. 3, we study overdetermined polynomial systems
$(f_{0},f_{1},\dots,f_{N})$ in $\mathbb{K}[\boldsymbol{\bar{X},\bar{Y}}]$
where $(f_{1},\dots,f_{N})$ is a square bipartite bilinear system and $f_{0}$
is a multilinear polynomial. We consider different types of polynomials
$f_{0}$. The obvious choice for $f_{0}$ is to have the same structure as one
of the polynomials $f_{1},\dots,f_{N}$; still we also choose $f_{0}$ to have a
different support. This leads to resultants of smaller degrees and so to
matrices of smaller size. The following $f_{0}$ lead to determinantal
formulas:

  1. 1.

$f_{0}\in\mathbb{K}[\boldsymbol{X}_{i}]_{\boldsymbol{1}}$, for any
$i\in\\{1,\dots,A\\}$.

  2. 2.

$f_{0}\in\mathbb{K}[\boldsymbol{Y}_{j}]_{\boldsymbol{1}}$, for any
$j\in\\{1,\dots,B\\}$.

  3. 3.

$f_{0}\in\mathbb{K}[\boldsymbol{X}_{i},\boldsymbol{Y}_{j}]_{\boldsymbol{1}}$,
for any $i\in\\{1,\dots,A\\}$ and $j\in\\{1,\dots,B\\}$.

######  Theorem 4.4.

Consider a generic overdetermined system $\boldsymbol{F}=(F_{0},\dots,F_{N})$
in $\mathbb{Z}[\boldsymbol{u}][\boldsymbol{\bar{X},\bar{Y}}]$ of multidegrees
${\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{N}}$ (Def. 2.3), where
$(F_{1},\dots,F_{n})$ is a square bipartite bilinear system. Assume that for
each $i\in\\{1,\dots,A\\}$, $\sum_{j=1}^{B}\mathcal{E}_{i,j}\geq\alpha_{i}$
and for each $j\in\\{1,\dots,B\\}$,
$\sum_{i=1}^{A}\mathcal{E}_{i,j}\geq\beta_{j}$ (see Rem. 4.3), and $F_{0}$ is
a multilinear polynomial as detailed in the paragraph above of multidegree
$\boldsymbol{d}_{0}$.

The degree vector
$\boldsymbol{m}=(m_{\boldsymbol{X}_{1}},\dots,m_{\boldsymbol{X}_{A}},m_{\boldsymbol{Y}_{1}},\dots,m_{\boldsymbol{Y}_{B}})$
define by

$\displaystyle\left\\{\begin{array}[]{lcl}m_{\boldsymbol{X}_{i}}=\sum_{j=1}^{B}\mathcal{E}_{i,j}-\alpha_{i}+d_{0,\boldsymbol{X}_{i}}&&\text{for
}1\leq i\leq A\\\ m_{\boldsymbol{Y}_{j}}=-1&&\text{for }1\leq j\leq
B\end{array}\right.$

corresponds to a Koszul-type determinantal formula (Def. 2.14)

$K_{\bullet}(\boldsymbol{m}):0\rightarrow
K_{1,\sum_{j=1}^{B}\beta_{j}+1}(\boldsymbol{m})\xrightarrow{\delta_{1}(\boldsymbol{m})}K_{0,\sum_{j=1}^{B}\beta_{j}}(\boldsymbol{m})\rightarrow
0.$

For the sake of brevity, we present the proof of Thm. 4.4 in Sec. A.2 as it is
similar to the one of Thm. 3.7. Additionally, we present an example of the
determinantal formulas constructed in this section in Sec. A.3.

##  5 Solving the Multiparameter Eigenvalue Problem

We present an algorithm (and an example) for solving a nonsingular MEP. The
polynomial system associated to MEP, see (3), corresponds to a star
multilinear system (Def. 3.1), where $A=1$, $B=\alpha$ and
$\mathcal{E}_{j}=\beta_{j}+1$, for each $j\in[B]$. In particular, following
(3), the system is
$\boldsymbol{f}^{\mathrm{MEP}}:=(f_{1,0},\dots,f_{1,\beta_{1}},\dots,f_{\alpha,0},\dots,f_{\alpha,\beta_{\alpha}})$,
where
$f_{i,j}\in\mathbb{K}[\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{i-1},\boldsymbol{Y}_{i+1},\dots,\boldsymbol{Y}_{\alpha}]_{\boldsymbol{1}}$,
for $i\in[\alpha]$ and $j+1\in[\beta_{i}+1]$. The expected number of solutions
is $\prod_{j=1}^{\alpha}(\beta_{j}+1)$, see Lem. 3.10.

We introduce a linear form
$f_{0}\in\mathbb{K}[\boldsymbol{X}]_{\boldsymbol{1}}$ and consider the
Sylvester-type determinantal formula of Cor. 3.8. The map $\delta$ associated
to this formula is as follows,

$\displaystyle\begin{array}[]{r
c}\delta:K[\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{\alpha}]_{\boldsymbol{1}}\times\prod_{i=1}^{\alpha}\mathbb{K}[\boldsymbol{Y}_{1}\dots\boldsymbol{Y}_{i-1},\boldsymbol{Y}_{i+1}\dots\boldsymbol{Y}_{\alpha}]_{\boldsymbol{1}}\rightarrow&\mathbb{K}[\boldsymbol{X},\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{\alpha}]_{\boldsymbol{1}}\\\
(g_{0},g_{1,0},\dots,g_{1,\beta_{1}},\dots,g_{\alpha,0},\dots,g_{\alpha,\beta_{\alpha}})\mapsto&g_{0}\,f_{0}+\sum_{i=1}^{\alpha}\sum_{j=0}^{\beta_{i}}g_{i,j}f_{i,j}.\end{array}$
(50)

We fix a monomial basis for the domain and codomain of $\delta$ and we
construct a matrix $C$ associated to it. We arrange the rows and columns of
$C$ so that we can write it as $\left[\begin{smallmatrix}C_{1,1}&C_{1,2}\\\
C_{2,1}&C_{2,2}\end{smallmatrix}\right]$, such that

  * •

The submatrix
$\left[\begin{smallmatrix}C_{2,1}&C_{2,2}\end{smallmatrix}\right]$ corresponds
to the rows $\boldsymbol{Y}^{\theta}\,f_{0}$, for
$\boldsymbol{Y}^{\theta}\in\mathbb{K}[\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{\alpha}]_{\boldsymbol{1}}$.

  * •

The submatrix $\left[\begin{smallmatrix}C_{1,2}\\\
C_{2,2}\end{smallmatrix}\right]$ corresponds to the column associated to the
monomial $\boldsymbol{Y}^{\theta}\,X_{0}$.

  * •

If the $k$-th row of $C$ corresponds to $\boldsymbol{Y}^{\theta}\,f_{0}$, then
the $k$-th column corresponds to the monomial
$\boldsymbol{Y}^{\theta}\,x_{0}$.

We say that a MEP is _affine_ if $\boldsymbol{f}^{\mathrm{MEP}}$ is a zero-
dimensional system and for every solution the $x_{0}$-coordinate is not zero.
When $\boldsymbol{f}^{\mathrm{MEP}}$ has a finite number of solutions, we can
always assume that it is _affine_ by performing a structured linear change of
coordinates. When the MEP is affine, by [5, Prop. 4.5], the matrix $C_{1,1}$
is invertible. Moreover, by [5, Lem. 4.4], we have a one to one correspondence
between the eigenvalues of the MEP and the (classical) eigenvalues of the
Schur complement of $C_{2,2}$,
$\widetilde{C_{2,2}}:=C_{2,2}-C_{2,1}C^{-1}_{1,1}C_{1,2}$ : each eigenvalue of
$\widetilde{C_{2,2}}$ is the evaluation of $\frac{f_{0}}{x_{0}}$ at an
eigenvalue of the original MEP. Also, the right eigenspaces of
$\widetilde{C_{2,2}}$ correspond to the vector of monomials
$x_{0}\,\boldsymbol{Y}^{\alpha}$, for
$\boldsymbol{Y}^{\theta}\in\mathbb{K}[\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{\alpha}]_{\boldsymbol{1}}$,
evaluated at each solution of
$(f_{1,0},\dots,f_{1,\beta_{1}},\dots,f_{\alpha,0},\dots,f_{\alpha,\beta_{\alpha}})$
[21, Lem. 5.1]. Even more, if we multiply these eigenvectors by
$\bigl{[}\begin{smallmatrix}M_{1,1}^{-1}\cdot M_{2,1}\\\
I\end{smallmatrix}\bigr{]}$, then we recover a vector corresponding to the
evaluation of every monomial in
$\mathbb{K}[\boldsymbol{X},\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{\alpha}]_{\boldsymbol{1}}$
evaluated at the original solution. This information suffices to recover all
the coordinates of the solution.

######  Remark 5.1 (Multiplication map).

For an affine MEP, the matrix $\widetilde{C_{2,2}}$ corresponds to the
multiplication map of the rational function $\frac{f_{0}}{x_{0}}$ in the
quotient ring
$\mathbb{K}[\boldsymbol{X},\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{\alpha}]/\boldsymbol{f}^{\mathrm{MEP}}$
at multidegree $(0,1,\dots,1)\in\mathbb{N}^{\alpha+1}$, with respect to the
monomial basis
$\\{\boldsymbol{Y}^{\theta}\\}_{\boldsymbol{Y}^{\theta}\in\mathbb{K}[\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{\alpha}]_{\boldsymbol{1}}}$.

Algorithm 1
$\texttt{SolveMEP}(\\{\\{M^{(i,j)}\\}_{j\in[\beta_{i}+1]}\\}_{i\in[\alpha]})$

0: Affine MEP $\\{\\{M^{(i,j)}\\}_{j\in[\beta_{i}+1]}\\}_{i\in[\alpha]}$

1: $(f_{1,0},\dots,f_{\alpha,\beta_{\alpha}})\leftarrow$ Multilinear system
associated to $\\{\\{M^{(i,j)}\\}_{j\in[\beta_{i}+1]}\\}_{i\in[\alpha]}$ (Eq.
3).

2: $f_{0}\leftarrow$ Generic linear polynomial in
$\mathbb{K}[\boldsymbol{X}]_{1}$.

3: $\bigl{[}\begin{smallmatrix}C_{1,1}&C_{1,2}\\\
C_{2,1}&C_{2,2}\end{smallmatrix}\bigr{]}$ $\leftarrow$ Matrix corresponding to
$\delta$ (Eq. 50); partitioned in four block

4:
$\left\\{\left(\frac{f_{0}}{x_{0}}(p),\bar{v}_{p}\right)\right\\}_{p}\leftarrow$
Set of Eigenvalue-Eigenvector of the Schur compl. of $C_{2,2}$.

5: for all
$\left(\frac{f_{0}}{x_{0}}(p),\bar{v}_{p}\right)\in\left\\{\left(\frac{f_{0}}{x_{0}}(p),\bar{v}_{p}\right)\right\\}_{p}$
do

6: Extract the coordinates of $p$ from
$\bigl{[}\begin{smallmatrix}C_{1,1}^{-1}\cdot C_{2,1}\\\
I\end{smallmatrix}\bigr{]}\cdot\bar{v}_{p}$.

######  Remark 5.2 (Atkinson’s Delta Method).

By inspecting the eigenvalues and eigenvectors of the Schur complement of
$C_{2,2}$ for $f_{0}=\frac{x_{i}}{x_{0}}$, we can conclude that
$\widetilde{C_{2,2}}$ equals the matrix $\Delta^{-1}_{0}\,\Delta_{i}$ from
Atkinson’s Delta method, see [3, Chp. 6]. It worth to point out that our
construction of $\widetilde{C_{2,2}}$ improves Atkinson’s construction of
$\Delta^{-1}_{0}\,\Delta_{i}$ as we avoid the symbolic expansion by minors
that he considers [3, Eq. 6.4.4]. The dimension of the matrix $C_{2,2}$, and
so the dimension of Atkinson’s Delta matrices, is
$\prod_{i=1}^{\alpha}(\beta_{i}+1)\times\prod_{i=1}^{\alpha}(\beta_{i}+1)$.
These matrices are dense. In contrast, the dimension of the matrix $C$ is
$(\alpha+1)\,\prod_{i=1}^{\alpha}(\beta_{i}+1)\times(\alpha+1)\,\prod_{i=1}^{\alpha}(\beta_{i}+1)$
(the degree of the resultant of the system), but the matrix is structured
(i.e., multi-Hankel matrix) and sparse; it has at most
$(\sum_{i=1}^{\alpha}\beta_{i}+\alpha+1)\,(\alpha+1)\,\prod_{i=1}^{\alpha}(\beta_{i}+1)$
non-zero positions.

In what follows we present an example of our algorithm in the two-parameter
eigenvalue problem (2EP). The interested reader can found information about
the applications of 2EP in physics in [29]. We consider the 2EP given by the
matrices

$\displaystyle\begin{array}[]{c c
c}M^{(1,0)}:=\left[\begin{array}[]{cc}-7&-3\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr-8&-2\end{array}\right]&M^{(1,1)}:=\left[\begin{array}[]{cc}12&2\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
13&1\end{array}\right]&M^{(1,2)}:=\left[\begin{array}[]{cc}-7&-1\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr-7&-1\end{array}\right]\\\\[20.0pt]
M^{(2,0)}:=\left[\begin{array}[]{cc}-11&-3\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 4&1\end{array}\right]&M^{(2,1)}:=\left[\begin{array}[]{cc}7&-1\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
1&2\end{array}\right]&M^{(2,1)}:=\left[\begin{array}[]{cc}-4&0\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr-1&-1\end{array}\right]\end{array}.$ (66)

For simplicity, we will name the three blocks of variables as
$\boldsymbol{X},\boldsymbol{Y},\boldsymbol{Z}$, instead of
$\boldsymbol{X}_{1},\boldsymbol{Y}_{1},\boldsymbol{Y}_{2}$. Following (3), we
write 2EP as the following bilinear system

$\begin{array}[]{l c l}\left[\begin{array}[]{c}f_{1}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr
f_{2}\end{array}\right]=&\left[\begin{array}[]{cc}-7\,x_{{0}}+12\,x_{{1}}-7\,x_{{2}}&-3\,x_{{0}}+2\,x_{{1}}-x_{{2}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr-8\,x_{{0}}+13\,x_{{1}}-7\,x_{{2}}&-2\,x_{{0}}+x_{{1}}-x_{{2}}\end{array}\right]&\cdot\left[\begin{array}[]{c}y_{0}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr y_{1}\end{array}\right]\\\\[20.0pt]
\left[\begin{array}[]{c}f_{3}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
f_{4}\end{array}\right]=&\left[\begin{array}[]{cc}-11\,x_{{0}}+7\,x_{{1}}-4\,x_{{2}}&-3\,x_{{0}}-x_{{1}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
4\,x_{{0}}+x_{{1}}-x_{{2}}&x_{{0}}+2\,x_{{1}}-x_{{2}}\end{array}\right]&\cdot\left[\begin{array}[]{c}z_{0}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr z_{{1}}\end{array}\right]\end{array}.$

According to Lem. 3.10, the 2EP should have $4$ different solutions. To solve
this system, we introduce a linear polynomial
$f_{0}\in\mathbb{K}[\boldsymbol{X}]$ that separates the eigenvalues of the
2EP, that is, if $\boldsymbol{\lambda}_{1}$ and $\boldsymbol{\lambda}_{2}$ are
different eigenvalues, then
$\frac{f_{0}}{x_{i}}(\boldsymbol{\lambda}_{1})\neq\frac{f_{0}}{x_{i}}(\boldsymbol{\lambda}_{2})$,
for some ${x_{i}}\in\boldsymbol{X}$. Then, we consider a Sylvester-like
determinantal formula for the resultant of $(f_{0},\dots,f_{4})$ (Cor. 3.8)
and we solve the original system using eigenvalue and eigenvector computations
as in [5].

If the MEP problem has a finite number of eigenvalues and all of them are
different, then any generic $f_{0}\in\mathbb{K}[\boldsymbol{X}]$ separates the
eigenvalues. In our case, we choose $f_{0}:=-x_{0}+5\,x_{1}-3\,x_{2}$.
Following Cor. 3.8, there is a Sylvester-type formula for the resultant of the
system $\boldsymbol{f}:=(f_{0},\dots,f_{4})$ using the degree vector
$\boldsymbol{m}:=(1,1,1)$. The latter is related to the determinantal data
$(\\{1,2\\},\emptyset,0)$. The Weyman complex reduces to

$\displaystyle 0\rightarrow\left(\begin{array}[]{l
}\phantom{\oplus\,}S_{X}(0)\otimes S_{Y}(1)\otimes
S_{Z}(1)\otimes\\{e_{0}\\}\\\ \oplus\,S_{X}(0)\otimes S_{Y}(0)\otimes
S_{Z}(1)\otimes\\{e_{1}\oplus e_{2}\\}\\\ \oplus\,S_{X}(0)\otimes
S_{Y}(1)\otimes S_{Z}(0)\otimes\\{e_{3}\oplus
e_{4}\\}\end{array}\right)\xrightarrow{\delta_{1}(\boldsymbol{m},\boldsymbol{f})}\left(S_{X}(1)\otimes
S_{Y}(1)\otimes S_{Z}(1)\otimes\mathbb{K}\right)\rightarrow 0,$

where the map $\delta_{1}(\boldsymbol{m},\boldsymbol{f})$ is a Sylvester map
(Prop. 2.17). Hence, the resultant of $\boldsymbol{f}$ is the determinant of a
matrix $C$ representing this map, which has dimensions $12\times 12$ (Case 1,
Lem. 3.11). We split $C$ in $\left[\begin{smallmatrix}C_{1,1}&C_{1,2}\\\
C_{2,1}&C_{2,2}\end{smallmatrix}\right]$ according to [5, Def. 4.1].

$\left[\begin{array}[]{c | cccccccc | cccc}&x_{{2}}y_{{0}}z_{{0}}&x_{{2}}y_{{0}}z_{{1}}&x_{{2}}y_{{1}}z_{{0}}&x_{{2}}y_{{1}}z_{{1}}&x_{{1}}y_{{0}}z_{{0}}&x_{{1}}y_{{0}}z_{{1}}&x_{{1}}y_{{1}}z_{{0}}&x_{{1}}y_{{1}}z_{{1}}&x_{{0}}y_{{0}}z_{{0}}&x_{{0}}y_{{0}}z_{{1}}&x_{{0}}y_{{1}}z_{{0}}&x_{{0}}y_{{1}}z_{{1}}\\\ \hline\cr z_{{0}}e_{1}&-7&&-1&&12&&2&&-7&&-3&\\\ z_{{1}}e_{1}&&-7&&-1&&12&&2&&-7&&-3\\\ z_{{0}}e_{2}&-7&&-1&&13&&1&&-8&&-2&\\\ z_{{1}}e_{2}&&-7&&-1&&13&&1&&-8&&-2\\\ y_{{0}}e_{3}&-4&0&&&7&-1&&&-11&-3&&\\\ y_{{1}}e_{3}&&&-4&0&&&7&-1&&&-11&-3\\\ y_{{0}}e_{4}&-1&-1&&&1&2&&&4&1&&\\\ y_{{1}}e_{4}&&&-1&-1&&&1&2&&&4&1\\\ \hline\cr y_{{0}}z_{{0}}e_{0}&-3&&&&5&&&&-1&&&\\\ y_{{0}}z_{{1}}e_{0}&&-3&&&&5&&&&-1&&\\\ y_{{1}}z_{{0}}e_{0}&&&-3&&&&5&&&&-1&\\\ y_{{1}}z_{{1}}e_{0}&&&&-3&&&&5&&&&-1\end{array}\right].$

######  Remark 5.3.

If the original MEP has a finite number of eigenvalues, after performing a
generic linear change of coordinates in the variables $\boldsymbol{X}$, we can
assume that there is no solution of $(f_{1},\dots,f_{n})$ such that $x_{0}=0$.

$\widetilde{C_{2,2}}=\left[\begin{array}[]{rrrr}\frac{7}{4}&0&-\frac{1}{4}&-\frac{1}{2}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr-\frac{3}{4}&\frac{3}{2}&\frac{9}{4}&2\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr-{\frac{21}{4}}&-3&{\frac{27}{4}}&\frac{5}{2}\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt\cr{\frac{69}{4}}&\frac{19}{2}&-{\frac{63}{4}}&-6\end{array}\right].$

By [5, Prop. 4.5], as the system $(f_{1},\dots,f_{n})$ has no solutions such
that $x_{0}=0$, the matrix $C_{1,1}$ is nonsingular. Hence, by [5, Lem. 4.4],
we have a one to one correspondence between the eigenvalues of the 2EP and the
(classical) eigenvalues of the Schur complement of $C_{2,2}$,
$\widetilde{C_{2,2}}:=C_{2,2}-C_{2,1}C^{-1}_{1,1}C_{1,2}$ : each eigenvalue of
$\widetilde{C_{2,2}}$ is the evaluation

of $\frac{f_{0}}{x_{0}}$ at an eigenvalue of the original 2EP. In our case
$\widetilde{C_{2,2}}$ is as it appears at the left.

Let $p_{1},\dots,p_{4}$ be the four solutions of $(f_{1},\dots,f_{4})$. Then,
the eigenvalues of $\widetilde{C_{2,2}}$ are
$\frac{f_{0}}{x_{0}}(p_{1})=1,\frac{f_{0}}{x_{0}}(p_{2})=2,\frac{f_{0}}{x_{0}}(p_{3})=3$
and $\frac{f_{0}}{x_{0}}(p_{4})=-2$. As
$\delta_{1}(\boldsymbol{m},\boldsymbol{f})$ is a Sylvester-like map, the right
eigenspaces of $\widetilde{C_{2,2}}$ contain the vector of monomials
$\boldsymbol{v}:=\left[\begin{smallmatrix}x_{{0}}y_{{0}}z_{{0}}\\\
x_{{0}}y_{{0}}z_{{1}}\\\ x_{{0}}y_{{1}}z_{{0}}\\\
x_{{0}}y_{{1}}z_{{1}}\end{smallmatrix}\right]$ evaluated at each solutions of
$(f_{1},\dots,f_{4})$ [21, Lem. 5.1]; this is as it appears in the table
below. If each eigenspace has dimension one, then we can recover some
coordinates of the solutions by inverting the monomial map given by
$\boldsymbol{v}$.

$\begin{array}[]{|
c|r|r|r|r|}\hline\cr\boldsymbol{v}&p_{1}&p_{2}&p_{3}&p_{4}\\\ \hline\cr
x_{{0}}y_{{0}}z_{{0}}&1&1&1&1\\\ x_{{0}}y_{{0}}z_{{1}}&-3&-1&-2&-3\\\
x_{{0}}y_{{1}}z_{{0}}&1&1&-1&-3\\\ x_{{0}}y_{{1}}z_{{1}}&-3&-1&2&9\\\
\hline\cr\end{array}$

For example, the $z_{0}$-coordinate of $p_{1}$ is non-zero as
$(x_{0}\,y_{0}\,z_{0})(p_{1})\neq 0$, and so its $z_{1}$-coordinate equals
$\frac{(x_{0}\,y_{0}\,z_{1})}{(x_{0}\,y_{0}\,z_{0})}(p_{1})=-3$. To compute
the remaining coordinates, either we substitute the computed coordinates of
the solutions in the original system and we solve a linear system, or we
extend each eigenvector $\boldsymbol{v}(p_{i})$ to $\boldsymbol{w}(p_{i})$,
where $\boldsymbol{w}(p_{i})$ is the solution of the following linear system:

$\left[\begin{smallmatrix}C_{1,1}&C_{1,2}\\\
C_{2,1}&C_{2,2}\end{smallmatrix}\right]\boldsymbol{w}(p_{i})=\frac{f_{0}}{x_{0}}(p_{i})\left[\begin{smallmatrix}\\\\[1.5pt]
0\\\\[3.0pt] \boldsymbol{v}(p_{i})\\\\[1.5pt] \end{smallmatrix}\right],\text{
and so }\boldsymbol{w}(p_{i})=\left[\begin{smallmatrix}-C_{1,1}^{-1}\cdot
C_{1,2}\,\boldsymbol{v}(p_{i})\\\
\boldsymbol{v}(p_{i})\end{smallmatrix}\right].$

Each coordinate of $\boldsymbol{w}(p_{i})$ is a monomial in
$\mathbb{K}[\boldsymbol{X}]_{1}\otimes\mathbb{K}[\boldsymbol{Y}]_{1}\otimes\mathbb{K}[\boldsymbol{Z}]_{1}$
evaluated at $p_{i}$. Hence, we can recover the coordinates of $p_{i}$ from
$\boldsymbol{w_{i}}(p_{i})$ by inverting a monomial map. In this case, the
solutions to $(f_{1},\dots,f_{4})$, and so eigenvalues and eigenvectors of 2EP
are

$\displaystyle\begin{array}[]{r c | c | c l}&\text{Ext. Eigenvalues}&\hbox to0.0pt{\text{\, Eigenvectors}\hss}&&\\\ &x_{0},x_{1},x_{2}&y_{0},y_{1}&z_{0},z_{1}&\\\ \hline\cr p_{1}=(&1,-1,-3&1,1&1,-3&)\\\ p_{2}=(&1,3,4&1,1&1,-1&)\\\ p_{3}=(&1,1,1&1,-1&1,-2&)\\\ p_{4}=(&1,1,2&1,-3&1,-3&)\end{array}$

## Acknowledgments

We thank Laurent Busé, Carlos D’Andrea, and Agnes Szanto for helpful
discussions and references, Jose Israel Rodriguez for pointing out the
relation between our paper [5] and MEP, and the anonymous reviewers for their
helpful comments. The authors are partially supported by ANR JCJC GALOP
(ANR-17-CE40-0009), the PGMO grant ALMA and the PHC GRAPE. M. R. Bender is
supported by ERC under the European’s Horizon 2020 research and innovation
programme (grant No 787840).

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##  Appendix A Additional proofs and examples

###  A.1 Proof of Lemma 3.11

###### Proof of Lemma 3.11.

Following Cor. 2.16, the size of the Koszul determinantal matrix is the degree
of the resultant. By Prop. 2.4, it holds,

$\text{degree}(\mathtt{res})=\sum_{k=0}^{N}\mathtt{MHB}({\boldsymbol{d}_{0}},\dots,{\boldsymbol{d}_{k-1}},{\boldsymbol{d}_{k+1}},\dots,{\boldsymbol{d}_{N}}).$

As many multidegrees $\boldsymbol{d}_{k}$ are identical, we couple the
summands in the previous equation. For each $j\in\\{1,\dots,B\\}$, let
$I_{j}\in\\{1,\dots,N\\}$ be the index of a polynomial in $\boldsymbol{F}$
such that
$F_{j}\in\mathbb{K}[\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{A},\boldsymbol{Y}_{I_{j}}]$.
Recall that $\mathcal{E}_{j}$ is the number of polynomials with multidegree
equal to $\boldsymbol{d_{I_{j}}}$. Hence, we rewrite the degree of the
resultant as

$\deg(\mathrm{Res}_{\mathcal{P}}(\boldsymbol{d}_{0},\dots,\boldsymbol{d}_{n}))=\mathtt{MHB}(\boldsymbol{d}_{1},\dots,\boldsymbol{d}_{n})+\sum_{j=1}^{B}\mathcal{E}_{j}\;\mathtt{MHB}(\boldsymbol{d}_{0},\dots,\boldsymbol{d}_{I_{j}-1},\boldsymbol{d}_{I_{j}+1},\dots,\boldsymbol{d}_{n}).$

From Lem. 3.10,
$\mathtt{MHB}(\boldsymbol{d}_{1},\dots,\boldsymbol{d}_{n})=\frac{(\sum_{i=1}^{A}\alpha_{i})!}{\prod_{i=1}^{A}\alpha_{i}!}\cdot\prod_{j=1}^{B}{\mathcal{E}_{j}\choose\beta_{j}}=\Upsilon$.
By Prop. 2.2, for every $1\leq j\leq B$,
$\mathtt{MHB}(\boldsymbol{d}_{0},\dots,\boldsymbol{d}_{I_{j}-1},\boldsymbol{d}_{I_{j}+1},\dots,\boldsymbol{d}_{n})$
is the coefficient of
$(\prod_{i=1}^{A}Z_{X_{i}}^{\alpha_{i}})(\prod_{t=1}^{B}Z_{Y_{t}}^{\beta_{t}})$
in

$\Big{(}\sum_{i=1}^{A}d_{0,\boldsymbol{X}_{i}}Z_{X_{i}}+\sum_{t=1}^{B}d_{0,\boldsymbol{Y}_{t}}Z_{X_{t}}\Big{)}\Big{(}\sum_{i=1}^{A}Z_{X_{i}}+Z_{Y_{j}}\Big{)}^{\mathcal{E}_{j}-1}\prod_{k\in\\{1,\dots,B\\}\setminus\\{j\\}}\Big{(}\sum_{i=1}^{A}Z_{X_{i}}+Z_{Y_{k}}\Big{)}^{\mathcal{E}_{k}}.$

Consider the last two factors of the previous equation, that is

$\displaystyle\Big{(}\sum_{i=1}^{A}Z_{X_{i}}+Z_{Y_{j}}\Big{)}^{\mathcal{E}_{j}-1}\prod_{k\in\\{1,\dots,B\\}\setminus\\{j\\}}\Big{(}\sum_{i=1}^{A}Z_{X_{i}}+Z_{Y_{k}}\Big{)}^{\mathcal{E}_{k}}.$
(67)

Then

$\mathtt{MHB}(\boldsymbol{d}_{0},\dots,\boldsymbol{d}_{I_{j}-1},\boldsymbol{d}_{I_{j}+1},\dots,\boldsymbol{d}_{n})=\sum_{s=1}^{A}d_{0,\boldsymbol{X}_{s}}\theta_{j,s}^{X}+\sum_{l=1}^{B}d_{0,\boldsymbol{Y}_{l}}\theta_{j,l}^{Y},$

where $\theta_{j,s}^{X}$ is the coefficient of
$\frac{(\prod_{i=1}^{A}Z_{X_{i}}^{\alpha_{i}})(\prod_{t=1}^{B}Z_{Y_{t}}^{\beta_{t}})}{Z_{X_{s}}}$
in (67), and $\theta_{j,t}^{Y}$ is the coefficient of
$\frac{(\prod_{i=1}^{A}Z_{X_{i}}^{\alpha_{i}})(\prod_{l=1}^{B}Z_{Y_{l}}^{\beta_{t}})}{Z_{Y_{t}}}$
in (67). After some computations, we have

$\displaystyle\theta_{j,s}^{X}=\frac{((\sum_{i=1}^{A}\alpha_{i})-1)!}{(\alpha_{s}-1)!\prod_{i\in\\{1,\dots,A\\}\setminus\\{s\\}}\alpha_{i}!}{\mathcal{E}_{j}-1\choose\beta_{j}}\prod_{k\in\\{1,\dots,B\\}\setminus\\{j\\}}{\mathcal{E}_{k}\choose\beta_{k}}=\Upsilon\cdot\frac{\alpha_{s}}{\sum_{i=1}^{A}\alpha_{i}}\frac{\mathcal{E}_{j}-\beta_{j}}{\mathcal{E}_{j}},$
$\displaystyle\theta_{j,t}^{Y}=\left\\{\begin{array}[]{ll}\frac{(\sum_{i=1}^{A}\alpha_{i})!}{\prod_{i=1}^{A}\alpha_{i}!}{\mathcal{E}_{j}-1\choose\beta_{j}-1}\prod_{k\in\\{1,\dots,B\\}\setminus\\{j\\}}{\mathcal{E}_{k}\choose\beta_{k}}=\Upsilon\cdot\frac{\beta_{j}}{\mathcal{E}_{j}}&\text{
if }t=j,\\\
\frac{(\sum_{i=1}^{A}\alpha_{i})!}{\prod_{i=1}^{A}\alpha_{i}!}{\mathcal{E}_{t}\choose\beta_{t}-1}{\mathcal{E}_{j}-1\choose\beta_{j}}\prod_{k\in\\{1,\dots,B\\}\setminus\\{t,j\\}}{\mathcal{E}_{k}\choose\beta_{k}}=\Upsilon\cdot\frac{\beta_{t}}{\mathcal{E}_{t}-\beta_{t}+1}\frac{\mathcal{E}_{j}-\beta_{j}}{\mathcal{E}_{j}}&\text{
otherwise.}\\\ \end{array}\right.$

Using the formulas for $\theta_{j,s}^{X}$ and $\theta_{j,t}^{Y}$ we get

$\displaystyle\deg(\mathrm{Res}_{\mathcal{P}}(\boldsymbol{d}_{0},\dots,\boldsymbol{d}_{n}))=\mathtt{MHB}(\boldsymbol{d}_{1},\dots,\boldsymbol{d}_{n})+\sum_{j=1}^{B}\mathcal{E}_{j}\;\mathtt{MHB}(\boldsymbol{d}_{0},\dots,\boldsymbol{d}_{I_{j}-1},\boldsymbol{d}_{I_{j}+1},\dots,\boldsymbol{d}_{n})=$
$\displaystyle\Upsilon+\sum_{j=1}^{B}\mathcal{E}_{j}\;(\sum_{s=1}^{A}d_{0,\boldsymbol{X}_{s}}\theta_{j,s}^{X}+\sum_{l=1}^{B}d_{0,\boldsymbol{Y}_{t}}\theta_{j,t}^{Y})=\Upsilon+\sum_{j=1}^{B}\mathcal{E}_{j}\;(\sum_{s=1}^{A}d_{0,\boldsymbol{X}_{s}}\theta_{j,s}^{X})+\sum_{j=1}^{B}\mathcal{E}_{j}\;(\sum_{l=1}^{B}d_{0,\boldsymbol{Y}_{t}}\theta_{j,t}^{Y}).$

Next, we simplify the last two summands of the previous equation. For the
first one, as