arXiv:1709.06624v1 [math.AG] 19 Sep 2017

On the multiplicity of isolated roots of sparse polynomial
systems∗
Marı́a Isabel Herrero♯,⋄ , Gabriela Jeronimo♯,†,⋄ , Juan Sabia†,⋄
♯ Departamento de Matemática, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Ciudad Universitaria, (1428) Buenos Aires, Argentina
† Departamento de Ciencias Exactas, Ciclo Básico Común,
Universidad de Buenos Aires, Ciudad Universitaria, (1428) Buenos Aires, Argentina
⋄ IMAS, UBA-CONICET, Buenos Aires, Argentina

September 21, 2017

Abstract
We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First,
we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these
formulas to solve the problem in the general case, by showing that the multiplicity
of an arbitrary affine isolated zero of a generic system with given supports equals the
multiplicity of the origin as a common zero of a generic system with an associated
family of supports.
The formulas obtained are in the spirit of the classical Bernstein’s theorem, in
the sense that they depend on the combinatorial structure of the system, namely,
geometric numerical invariants associated to the supports, such as mixed volumes of
convex sets and, alternatively, mixed integrals of convex functions.

1

Introduction

The connections between the set of solutions of a polynomial system and the geometry
of the supports of the polynomials involved have been vastly studied in the literature,
starting with the foundational work of Bernstein ([1]), Kushnirenko ([13]) and Khovanskii
([11]). They proved that the number of isolated solutions in (C∗ )n of a system with n
polynomial equations in n unknowns is bounded by the mixed volume of their support sets.
Afterwards, combinatorial invariants of the same type also allowed to obtain bounds for
the number of isolated solutions of the system in the affine space Cn (see, for example, [20],
[14], [9] and [5]). In [19], another refinement of Bernstein’s bound was given by introducing
mixed integrals of concave functions to estimate the number of isolated solutions in C ×
∗

Partially supported by the following Argentinian grants: PIP 11220130100527CO CONICET (20142016) and UBACYT 2017, 20020160100039BA.

1

(C∗ )n−1 . In addition, the equidimensional decomposition of the affine variety defined by
a generic system with given supports and the degree of this variety are also related to the
geometry and combinatorics of the supports (see [8]).
Even though for generic sparse polynomial systems their common zeroes in (C∗ )n are
simple, their isolated roots with zero coordinates may have greater multiplicities. The aim
of this paper is to prove formulas for the multiplicity of the isolated affine zeroes of generic
sparse polynomial systems in terms of the geometry of their supports. This dependence
is already present in the seminal work of Kushnirenko ([12]), where the Milnor number of
the singularity at the origin of a hypersurface is studied.
Several authors have used geometric tools, including convex sets, volumes and covolumes, to solve related problems. Geometric invariants of this type are considered in [23] to
determine multiplicities of monomial ideals in local rings. In [6, Chapter 5], the multiplicity of a singular point on a toric variety is given as a normalized volume. A particular case
of this result is recovered in [3], where the multiplicity of the origin as an isolated zero of a
generic unmixed polynomial system is computed under the assumption that each polynomial contains a pure power of each variable. A generalization of this result to the mixed
case under the same assumption can be found in [10], where the multiplicity of the origin
is expressed in terms of mixed covolumes. Recently, in [16] a formula for the intersection
multiplicity at the origin of the hypersurfaces defined in Cn by n generic polynomials with
fixed Newton diagrams is proved.
In this paper, we obtain formulas for the multiplicities of all the affine isolated zeros
of a generic polynomial system of n polynomials in n variables with given supports in
terms of mixed volumes and, alternatively, in terms of mixed integrals of convex functions
associated to the supports of the polynomials involved.
First, we consider the case of the origin. In this case, our formulas for the multiplicity
can be seen as a generalization of those in [10], in the sense that the only hypotheses on the
supports we make are the necessary ones, proved in [8, Proposition 6], so that the origin
is an isolated zero of a generic system with the given supports. The previous approach
from [16] to compute the multiplicity of the origin under no further assumptions on the
supports leads to a formula which, unlike ours, is not symmetric in the input polynomials.
In order to analyze the case of arbitrary affine isolated zeros, the result in [8, Proposition 6] enables us to determine all sets I ⊂ {1, . . . , n} such that a generic system with
the given supports has isolated zeros whose vanishing coordinates are indexed by I. For
such an isolated zero, we prove that its multiplicity equals the multiplicity of the origin as
an isolated zero of an associated generic sparse system of #I polynomials in #I variables
whose supports can be explicitly defined from the input supports and the set I. Thus,
a formula for the multiplicity of an arbitrary affine zero of the system follows from our
previous result concerning the multiplicity of the origin.
The paper is organized as follows: Section 2 recalls the definitions and basic properties
of mixed volumes and mixed integrals, and describes the algorithmic approach to compute
multiplicities of isolated zeros of polynomial systems by means of basic linear algebra given
in [4], which we use as a tool. In Section 3, formulas for the multiplicity of the origin are
obtained, first for systems where each polynomial contains a pure power of each variable
and then, in the general case. Finally, Section 4 is devoted to computing the multiplicity
of an arbitrary affine isolated zero of a generic system.

2

2

Preliminaries

2.1

Mixed volume and stable mixed volume

Let A1 , . . . , An be finite subsets of (Z≥0 )n . A sparse polynomial system supported on
A = (A1 , . . . , An ) is given by polynomials
X
fj =
cj,a xa
a∈Aj

in the variables x = (x1 , . . . , xn ), with cj,a ∈ C \ {0} for each a ∈ Aj and 1 ≤ j ≤ n.
We denote by M Vn (A) = M Vn (A1 , . . . , An ) the mixed volume of the convex hulls of
A1 , . . . , An in Rn , which is defined as
X

X
conv(Aj )
M Vn (A) =
(−1)n−#J V oln
j∈J

J⊂{1,...,n}

(see, for example, [2, Chapter 7]). The mixed volume of A is an upper bound for the
number of isolated roots in (C∗ )n of a sparse system supported on A (see [1]).
The stable mixed volume of A = (A1 , . . . , An ), denoted by SMn (A) = SMn (A1 , . . . , An ),
is introduced in [9] to estimate the number of isolated roots in Cn of a sparse polynomial
system supported on A and is defined as follows. Let A0 = (A01 , . . . , A0n ) be the family
with A0j := Aj ∪ {0} for every 1 ≤ j ≤ n, and let ω 0 = (ω10 , . . . , ωn0 ) be the lifting function
for A0 defined by ωj0 (q) = 0 if q ∈ Aj and ωj0 (0) = 1 if 0 ∈
/ Aj . Consider the polytope
0
n+1
Q in R
obtained by taking the Minkowski (pointwise) sum of the convex hulls of the
graphs of ω10 , . . . , ωn0 . The projection of the lower facets of Q0 (that is, the n-dimensional
faces with inner normal vector with a positive last coordinate) induces a subdivision of
A0 . A cell C = (C1 , . . . , Cn ), with Cj ⊂ A0j for every 1 ≤ j ≤ n, of this subdivision is
said to be stable if it corresponds to a facet of Q0 having an inner normal vector with all
non-negative coordinates. The stable mixed volume SMn (A1 , . . . , An ) is the sum of the
mixed volumes of all the stable cells in the subdivision of A0 .
Note that A = (A1 , . . . , An ) is a stable cell in the defined subdivision of A0 , namely,
the cell with associated inner normal vector (0, . . . , 0, 1); therefore, we have that
M Vn (A1 , . . . , An ) ≤ SMn (A1 , . . . , An ) ≤ M Vn (A1 ∪ {0}, . . . , An ∪ {0}).

2.2

Mixed integrals for concave and convex functions

Let P1 , . . . , Pn be polytopes in Rn−1 , and, for 1 ≤ j ≤ n, let σj : Pj → R be a concave
function and ρj : Pj → R a convex function. Following [18], we can define concave
(respectively convex) functions as:
σi ⊞ σj : Pi + Pj → R,

σi ⊞ σj (x) = max{σi (y) + σj (z) : y ∈ Pi , z ∈ Pj , y + z = x}

ρi ⊞′ ρj : Pi + Pj → R,

ρi ⊞′ ρj (x) = min{ρi (y) + ρj (z) : y ∈ Pi , z ∈ Pj , y + z = x}.

Note that ρi ⊞′ ρj = −(−ρi ) ⊞ (−ρj ).
In the same way, for every non-empty subset J ⊂ {1, . . . , n}, we can define
X
X
⊞j∈J σj :
Pj → R and
⊞′j∈J ρj :
Pj → R.
j∈J

j∈J

3

The mixed integrals of σ1 , . . . , σn (respectively, ρ1 , . . . , ρn ) are defined as:

k=1

X

J ⊂{1,...,n}
#J =k

Z

n
X

X

Z

n
X
(−1)n−k
M In (σ1 , . . . , σn ) =

M In′ (ρ1 , . . . , ρn ) =

(−1)n−k

k=1

J ⊂{1,...,n}
#J =k

P

P

j∈J

j∈J

⊞j∈J σj (x)dx,
Pj

Pj

⊞′j∈J ρj (x)dx.

For a polytope P ⊂ Rn−1 , a convex function ρ : P → R and a concave function
σ : P → R such that ρ(x) ≤ σ(x) for every x ∈ P , we denote
Pρ,σ = conv({(x, ρ(x)) : x ∈ P } ∪ {(x, σ(x)) : x ∈ P }).
Given a polytope Q ⊂ Rn , if π : Rn → Rn−1 is the projection to the first n − 1
coordinates, we may define a concave function σQ : π(Q) → R and a convex function
ρQ : π(Q) → R as:
σQ (x) = max{xn ∈ R : (x, xn ) ∈ Q}

and

ρQ (x) = min{xn ∈ R : (x, xn ) ∈ Q}.

Remark 1 The functions σQ and ρQ defined above parameterize the lower and upper
envelopes of Q respectively. Moreover, π(Q)ρQ ,σQ = Q.
Let Q1 , . . . , Qn be polytopes in Rn . P
For 1 ≤ j ≤ n, let σj = σQj and
P ρj = ρQj . Let
′
)
→
R
and
⊞
ρ
:
J ⊂ {1, . . . , n}, J 6= ∅. Then, ⊞j∈J σj : j∈J π(Q
j
j
j∈J π(Qj ) → R
j∈J
P
parameterize the upper and lower envelopes of j∈J Qj respectively.

2.3

Multiplicity matrices

In order to compute multiplicities of isolated zeros of polynomial systems, we will follow
the algorithmic approach from [4] based on duality theory, which we briefly recall in this
section.
Let f = (f1 , . . . , fn ) be a system of polynomials in C[x1 , . . . , xn ]. Denote I the ideal
of C[x] = C[x1 , . . . , xn ] generated by f1 , . . . , fn .
For an isolated zero ζ ∈ Cn of the system f , we denote multζ (f ) its multiplicity,
defined as the dimension (as a C-vector space) of the local ring C[x]mζ /IC[x]mζ , where
mζ = (x1 − ζ1 , . . . , xn − ζn ) is the maximal ideal associated with ζ (see, for instance, [2,
Chapter 4, Definition (2.1)]).
Let Dζ (I) the dual space of the ideal I at ζ; namely, the vector space
n
o
X
Dζ (I) = c =
cα ∂α [ζ] | c(f ) = 0 for all f ∈ I ,
α∈(Z≥0 )n

where, for every α = (α1 , . . . , αn ) ∈ (Z≥0 )n , cα ∈ C,
∂α =

∂ |α|
1
,
α1
α1 ! . . . αn ! ∂x1 . . . , xαnn
4

(1)

and
∂α [ζ] : C[x] → C,

∂α [ζ](f ) = (∂α f )(ζ).

The dimension of Dζ (I) equals the multiplicity of ζ as a zero of I (see [15], [22]).
For every k ≥ 0, consider the subspace
n
o
X
cα ∂α [ζ] | c(f ) = 0 for all f ∈ I
Dζk (I) = c =
α∈(Z≥0 )n , |α|≤k

of all functionals in Dζ (I) with differential order bounded by k. Since ζ is an isolated
common zero of I, there exists k0 ∈ Z≥0 such that Dζ (I) = Dζk0 (I) = Dζk (I) for all k ≥ k0
and dim(Dζk (I)) < dim(Dζk+1 (I)) for every 0 ≤ k < k0 (see [4, Lemma 1]).
Following [4, Section 4], the dimension of the vector spaces Dζk (I) can be computed
by means of the multiplicity matrices, defined as follows. For k = 0, set S0 (f , ζ) =
[f1 (ξ) · · · fn (ξ)]t = 0 ∈ Cn×1 . Take ≺ a graded monomial ordering. For k ≥ 1, consider
the sets Ik = {α ∈ (Z≥0 )n | |α| ≤ k} ordered by ≺, and Ik−1 × {1, . . . , n} with the
 ordering
k+n
(β, j) ≺ (β ′ , j ′ ) if β ≺ β ′ or β = β ′ and j < j ′ . Let Sk (f , ζ) be the k−1+n
k−1 n ×
k
matrix whose columns are indexed by Ik (corresponding to the differential functionals ∂α
for α ∈ Ik ) and whose rows are indexed by (β, j) ∈ Ik−1 × {1, . . . , n} (corresponding to
the polynomials (x − ζ)β fj ) such that the entry at the intersection of the row indexed by
(β, j) and the column indexed by α is
(Sk (f , ζ))(β,j),α = ∂α ((x − ζ)β fj )(ζ).
(Here, (x − ζ)β = (x1 − ζ1 )β1 · · · (xn − ζn )βn .) Then, the dimension of Dζk (I) equals the
dimension of the nullspace of Sk (f , ζ) (see [4, Theorems 1 and 2]). As a consequence:
Proposition 2 With the previous assumptions and notation, if
k0 = min{k ∈ Z≥0 | dim(ker(Sk (f , ζ))) = dim(ker(Sk+1 (f , ζ)))},
the multiplicity of ζ as an isolated zero of f is multζ (f ) = dim(ker(Sk (f , ζ))) for any
k ≥ k0 .

3

Multiplicity of the origin

Consider a family A = (A1 , . . . , An ) of finite sets in (Z≥0 )n such that 0 6∈ Aj for all
1 ≤ j ≤ n. Under this assumption, 0 ∈ Cn is a common zero of any sparse system of
polynomials f1 , . . . , fn ∈ C[x1 , . . . , xn ] supported on A.
We are interested in the case when 0 is an isolated common zero of the system. By
[8, Proposition 6], for a generic family of polynomials f = f1 , . . . , fn ∈ C[x1 , . . . , xn ]
supported on A, we have that 0 is an isolated point of V (f ) if and only if #I + #JI ≥ n
for all I ⊂ {1, . . . , n}, where JI is the set of subindexes of all polynomials that do not
vanish when we evaluate xi = 0 for all i ∈ I.
Every c = (c1 , . . . , cn ) ∈ C#A1 × · · · × C#An defines a system fc of polynomials with
coefficients c supported on a family of subsets of A1 , . . . , An . If 0 is an isolated zero of fc ,
we define multA (c) := mult0 (fc ) ∈ Z>0 .
5

Lemma 3 Under the previous assumptions and notation, let µA be the minimum of the
function multA . Then, {c ∈ C#A1 × · · · × C#An | multA (c) = µA } contains a non-empty
Zariski open set of C#A1 × · · · × C#An .
Proof: It is straightforward, for example, from the computation of multiplicities by using
multiplicity matrices (see Section 2.3).

In this sense, we may speak of µA as the multiplicity of 0 as an isolated root of a
generic sparse system supported on A.
Therefore, in this section, we will focus on the computation of the multiplicity of the
origin as a common zero of a generic polynomial system supported on A = (A1 , . . . , An ),
under the following assumptions:
(H1) 0 ∈
/ Aj ⊂ (Z≥0 )n for every 1 ≤ j ≤ n;
(H2) for all I ⊂ {1, . . . , n}, if JI := {j ∈ {1, . . . , n} | ∃a ∈ Aj : ai = 0 ∀i ∈ I}, then
#I + #JI ≥ n.
Moreover, in [8, Proposition 5], these conditions are proved to be equivalent to the fact
that, for a generic system f supported on A and vanishing at 0 ∈ Cn , the variety V (f )
consists only of isolated points in Cn .
Under these assumptions, by [9, Theorem 2], the number of common zeros of f in Cn
counted with multiplicities is the stable mixed volume SMn (A). In particular, since the
number of common zeros of the system in (C∗ )n is the mixed volume M Vn (A) (see [1]),
we have that
mult0 (f ) ≤ SMn (A) − M Vn (A) ≤ M Vn (A0 ) − M Vn (A),

(2)

where A0 = (A1 ∪ {0}, . . . , An ∪ {0}).

3.1

A particular case

The first case we are going to consider is when the following stronger assumption on A
holds:
(H3) For every 1 ≤ i, j ≤ n, there exists µij ∈ N such that µij ei ∈ Aj , where ei is the ith
vector of the canonical basis of Qn .
Note that assumption (H3) implies that assumption (H2) holds.
Under condition (H3), in [10, Theorem 7.6] the multiplicity of the origin as an isolated
common zero of a generic polynomial system supported on A is computed in terms of
covolumes of coconvex bodies associated to A. Here, we will first re-obtain this result by
proving a formula using mixed volumes of convex polytopes and then, we will reformulate
this formula in terms of mixed integrals of convex functions.
We start by comparing stable mixed volumes with mixed volumes in our particular
setting.

6

Lemma 4 With the previous notation, if assumptions (H1) and (H3) hold, we have that
SMn (A1 , . . . , An ) = M Vn (A01 , . . . , A0n ).
Proof: It suffices to prove that every cell in the subdivision of A0 = (A01 , . . . , A0n ) induced
by the lifting function introduced in Section 2.1 is stable.
Consider a cell C = (C1 , . . . , Cn ) of the stated subdivision different from (A1 , . . . , An )
(for which the result is trivial), and let η = (η1 , . . . , ηn , 1) be its associated inner normal
vector. We have to show that ηi ≥ 0 for every 1 ≤ i ≤ n.
For every 1 ≤ j ≤ n, there exists aCj ∈ R such that aCj = η. (q, ωj0 (q)) for all q ∈ Cj
and aCj ≤ η. (q, ωj0 (q)) for all q ∈ A0j . As the cell C is not (A1 , . . . , An ), there exists j0
such that 0 ∈ Cj0 and 0 ∈
/ Aj0 ; then, aCj0 = η. (0, 1) = 1. Since, by assumption (H3), for
all 1 ≤ i ≤ n, there exists µij0 ∈ N such that µij0 ei ∈ A0j0 , then, 1 = aCj0 ≤ η . µij0 (ei , 0) =
ηi µij0 . The result follows from the fact that µij0 > 0 for all 1 ≤ i ≤ n.

Now, we can state our first formula for the multiplicity of the origin.
Proposition 5 Let f = (f1 , . . . , fn ) be a generic polynomial system in C[x1 , . . . , xn ] supported on a family A = (A1 , . . . , An ) of finite sets of (Z≥0 )n satisfying assumptions (H1)
and (H3). Then, the origin is an isolated common zero of f and
mult0 (f ) = M Vn (A0 ) − M Vn (A).
Proof: Assumption (H1) implies that the origin is a common zero of the polynomials f .
In addition, by assumption (H3), the only common zero of f not in (C∗ )n is the origin.
Then, all the common zeros of f in Cn are isolated and so, the number of these common
zeros is SMn (A) (see [9]). Finally, since the number of common zeros of f in (C∗ )n
is M Vn (A) (see [1]) and all these zeros have multiplicity 1 (see [17]), we deduce that
M Vn (A) + mult0 (f ) = SMn (A). Thus, the result follows from Lemma 4.

Example 1 Consider the generic polynomial system f = (f1 , f2 , f3 ) with
f1 = c11 x1 + c12 x2 + c13 x22 + c14 x21 x2 x3 + c15 x73
f2 = c21 x21 + c22 x31 + c23 x21 x2 + c24 x33 + c25 x72
f3 = c31 x1 + c32 x1 x2 + c33 x23 + c34 x2 x33 + c35 x72
with support family A = (A1 , A2 , A3 ), where
A1 = {(1, 0, 0), (0, 1, 0), (0, 2, 0), (2, 1, 1), (0, 0, 7)}
A2 = {(2, 0, 0), (3, 0, 0), (2, 1, 0), (0, 0, 3), (0, 7, 0)}
A3 = {(1, 0, 0), (1, 1, 0), (0, 0, 2), (0, 1, 3), (0, 7, 0)}
satisfying assumptions (H1) and (H3). Then, Proposition 5 states that 0 is an isolated
common root of f with multiplicity
mult0 (f ) = M V3 (A0 ) − M V3 (A) = 147 − 144 = 3.

7

In order to restate the formula in the previous proposition by means of a mixed integral
of suitable convex functions, we first introduce further notation and prove some auxiliary
results.
For 1 ≤ j ≤ n, let Qj = conv(Aj ) and ∆j = conv{0, λ1j e1 , . . . , λnj en }, where
λij = min{µ ∈ N | µei ∈ Qj }

for 1 ≤ i ≤ n.

(3)

Let π : Rn → Rn−1 be the projection to the first n − 1 coordinates. As in Section 2.2,
let σj : π(Qj ) → R denote the concave function that parameterizes the upper envelope of
Qj and ρj : π(Qj ) → R the convex function that parameterizes its lower envelope. Since
π(∆j ) ⊂ π(Qj ), we may consider
σ j = σj |π(∆j )

and

ρj = ρj |π(∆j ) ,

(4)

the restrictions of these functions to π(∆j ).
For a non-empty set J ⊂ {1, . . . , n}, we denote
X
X
∆J :=
∆j ,
QJ :=
Qj .
j∈J

j∈J

Lemma 6 Let J ⊂ {1, . . . , n} be a non-empty set. Then, every facet of ∆J that is not
contained in a hyperplane {xi = 0}, for 1 ≤ i ≤ n, has an inner normal vector with all
negative coordinates. We will call these facets the non-trivial facets of ∆J .
Proof: If J = {j} for some 1 ≤ j ≤ n, the result is straightforward because the only facet
satisfying the required conditions is F = conv{λ1j e1 , . . . , λnj en }, and λij ∈ N for every
1 ≤ i ≤ n.
Let F beP
a non-trivial facet of ∆J and η = (η1 , . . . , ηn ) an inner normal vector of F .
Then, F =
j∈J Fj , where Fj is a face of ∆j with inner normal vector η. For every
1 ≤ i ≤ n, since F is not contained in the hyperplane {xi = 0}, there exists ji ∈ J such
that λiji ei ∈ Fji ; then,
0 = η. 0 ≥ η. λiji ei = ηi λiji
(5)
and, so ηi ≤ 0.
If 0 ∈ Fj for some j ∈ J, then η. q ≥ η. 0 = 0 for every q ∈ ∆j ; in particular,
ηk λkj = η. λkj ek ≥ η. 0 = 0 for every 1 ≤ k ≤ n. This implies that η = 0, a contradiction.
Then, 0 ∈
/ Fj for every j ∈ J, the inequalities in (5) are strict and, therefore, ηi < 0 for
every 1 ≤ i ≤ n.

Lemma 7 Let J ⊂ {1, . . . , n} be a non-empty set. Then, for every point x in a non-trivial
facet of π(∆J ) we have that (⊞′j∈J ρj )(x) = 0.
Pn−1
Proof: If J =P
{j}, x ∈ conv{λ1j π(e1 ), . . . , λn−1,j π(en−1 )}, thatP
is, x = i=1
ti λij π(ei )
n−1
n−1
ti ρj (λij π(ei )) = 0.
for ti ≥ 0 with i=1
ti = 1. Then, since ρj is convex, 0 ≤ ρj (x) ≤ i=1
P
If #J > 1, let x be in a nontrivial facet F of P
π(∆J ). We have that F = j∈J Fj , with
Fj a face of π(∆j ) such that 0 ∈
/ Fj ; then, x = j∈J pj with pj ∈ F
j . Hence, ρj (pj ) = 0
P
′
′
and, by the definition of ⊞j∈J ρj , it follows that 0 ≤ (⊞j∈J ρj )(x) ≤ j∈J ρj (pj ) = 0. 
8

Lemma 8 For every non-empty subset J of {1, . . . , n}, the convex function ⊞′j∈J ρj defined
over π(∆J ) parameterizes the lower envelope of QJ over the points of π(∆J ).
Proof: For every J ⊂ {1, . . . , n}, we denote
X
X
PJ :=
π(Qj ), DJ :=
π(∆j )
j∈J

j∈J

and ⊞′j∈J ρj to the restriction of ⊞′j∈J ρj : PJ → R to DJ ⊂ PJ . With this notation, we
have to prove that
(6)
⊞′j∈J ρj = ⊞′j∈J ρj .
Before proceeding, we will state three basic results that will be applied throughout the
proof. We use the notation
ρJ := ⊞′j∈J ρj .
Claim I. If p1 lies in a non-trivial facet of DJ and p2 ∈ PJ , then for every x lying on the
line segment p1 p2 , we have ρJ (x) ≤ ρJ (p2 ): as x = (1−t)p1 +tp2 for 0 ≤ t ≤ 1, ρJ is convex
and ρJ ≡ 0 on the non-trivial facets of DJ , ρJ (x) ≤ (1 − t)ρJ (p1 ) + t ρJ (p2 ) = t ρJ (p2 ).
Claim II. If p1 ∈ DJ and p2 ∈
/ DJ , then for every x 6= p2 lying on the line segment p1 p2 ,
since DJ is a convex set, d(p1 , DJ ) < d(p2 , DJ ), where d(·, DJ ) is the distance to DJ .
Claim III. If p1 ∈ DJ and p2 ∈ (R≥0 )n−1 \DJ there exists t ∈ (0, 1] such that tp1 +(1−t)p2
lies in a non-trivial facet of DJ .
The proof will be done recursively. For a fixed non-empty set J ⊂ {1, . . . , n}, let J1 , J2
be disjoints sets such that J = J1 ∪ J2 and assume that identity (6) holds for each of them.
We will prove that if ρJk := ⊞′j∈Jk ρj , for k = 1, 2, then ρJ1 ⊞′ ρJ2 = ρJ1 ⊞′ ρJ2 .
Let x ∈ DJ . Then, there exist y0 ∈ DJ1 and z0 ∈ DJ2 such that x = y0 + z0 . Let
y ′ ∈ PJ1 and z ′ ∈ PJ2 be such that x = y ′ + z ′ and ρJ1 ⊞′ ρJ2 (x) = ρJ1 (y ′ ) + ρJ2 (z ′ ). If
y ′ ∈ DJ1 and z ′ ∈ DJ2 the result follows.
We first show that there exist y ′ and z ′ as before satisfying that y ′ ∈ DJ1 or z ′ ∈ DJ2 .
For every 0 ≤ t ≤ 1, if yt = (1 − t)y0 + ty ′ and zt = (1 − t)z0 + tz ′ , then x = yt + zt . If
y′ ∈
/ DJ1 and z ′ ∈
/ DJ2 , there exist 0 < t1 , t2 ≤ 1 such that yt1 and zt2 lie in non-trivial
facets of DJ1 and DJ2 respectively. Consider t0 = min{t1 , t2 }; then x = yt0 + zt0 and, by
Claim I, ρJ1 ⊞ ρJ2 (x) = ρJ1 (yt0 ) + ρJ2 (zt0 ).
Now, without loss of generality, assume that z ′ ∈ DJ2 . Consider the compact set
Cx = {y ∈ PJ1 | x − y ∈ DJ2 and ρJ1 ⊞′ ρJ2 (x) = ρJ1 (y) + ρJ2 (x − y)}.
We will prove that Cx ∩DJ1 6= ∅. If not, let y ∈ Cx be such that d(Cx , DJ1 ) = d(y, DJ1 ) > 0.
First, assume that z := x−y does not lie in a non-trivial facet of DJ2 . This implies that
z+w ∈ DJ2 for every w with sufficiently small non-negative coordinates. Let 0 < ǫ < 1 such
that (1−ǫ)y ∈
/ DJ1 and that z+ǫy ∈ DJ2 . Claims III and I imply that ρJ1 ((1−ǫ)y) ≤ ρJ1 (y)
and that ρJ2 (z + ǫy) ≤ ρJ2 (z) and, therefore, ρJ1 ⊞′ ρJ2 (x) = ρJ1 ((1 − ǫ)y) + ρJ2 (z + ǫy).
As, by Claim II, d((1 − ǫ)y, DJ1 ) < d(y, DJ1 ) we have a contradiction.
Assume now that z := x − y lies in non-trivial facets of DJ2 .
Recall that x = y0 + z0 with y0 ∈ DJ1 , z0 ∈ DJ2 . If z and z0 lie in the same non-trivial
facet of DJ2 , then the line segment zz0 is contained in this facet. On the other hand,
9

there exists 0 ≤ t ≤ 1 such that (1 − t)y0 + ty lies in a non-trivial facet of DJ1 . Therefore,
x = ((1−t)y0 +ty)+((1−t)z0 +tz), ρJ1 ⊞′ ρJ2 (x) = ρJ1 ((1−t)y0 +ty)+ρJ2 ((1−t)z0 +tz) = 0
and so, (1 − t)y0 + ty ∈ Cx ∩ DJ1 , which is a contradiction.
If z0 does not lie in any of the non-trivial facets of DJ2 containing z, let η 1 , . . . , η k
be inner normal vectors to these facets and consider the hyperplanes parallel to them
and containing y, which are defined by the equations η ℓ .(Y − y) = 0 for 1 ≤ ℓ ≤ k. As
η ℓ .y + η ℓ .z = η ℓ .y0 + η ℓ .z0 and η ℓ .z < η ℓ .z0 , then η ℓ .y0 < η ℓ .y. In addition, since all the
coordinates of η ℓ are negative (see Lemma 6) and y ∈ (R≥0 )n−1 , then η ℓ .y < 0. Therefore,
the hyperplane η ℓ .(Y −y) = 0 intersects the line segment 0y0 in a point λℓ y0 with 0 ≤ λℓ ≤
1. If λ = max{λℓ / 1 ≤ ℓ ≤ k}, consider yt = (1 − t)y + tλy0 and zt = x − yt for 0 ≤ t ≤ 1.
For t sufficiently small, we will show that zt ∈ DJ2 , that ρJ1 ⊞′ ρJ2 (x) = ρJ1 (yt ) + ρJ2 (zt )
and that d(yt , DJ1 ) < d(y, DJ1 ), which leads to a contradiction.
For 1 ≤ ℓ ≤ k, as λ ≥ λℓ , η ℓ .(y − λy0 ) ≥ 0; then η ℓ .(y − yt ) ≥ 0 and so η ℓ .zt =
ℓ
η .z + η ℓ .(y − yt) ≥ η ℓ .z. If z lies in a trivial facet of DJ2 , that is, zi = 0 for some 1 ≤ i ≤ n,
then yi = xi ; as (y0 )i ≤ xi , we have that (zt )i = t(yi − λ(y0 )i ) ≥ 0. Taking t sufficiently
small, zt satisfies all the remaining inequalities defining DJ2 and so, zt ∈ DJ2 . Moreover,
since y ∈
/ DJ1 , for t sufficiently small, yt ∈
/ DJ1 . Then, by Claim I, ρJ1 (yt ) ≤ ρJ1 (y). On
the other hand, zt lies in the same non-trivial facet of DJ2 as z, namely, the facet defined
by η ℓ0 . (Z − z) = 0 for ℓ0 such that λ = λℓ0 and, therefore, ρJ2 (zt ) = 0. We conclude that
ρJ1 (yt ) + ρJ2 (zt ) = ρJ1 ⊞′ ρJ2 (x). Finally, the inequality d(yt , DJ1 ) < d(y, DJ1 ) holds by
Claim II.

For every 1 ≤ j ≤ n, let Q0j = conv(Aj ∪{0}) and σj0 , ρ0j the functions that parameterize
its upper and lower envelopes respectively. Assumption (H3) ensures that π(Q0j ) = π(Qj ).
Lemma 9 For every 1 ≤ j ≤ n, ρ0j (x) =

(

0
ρj (x)

if x ∈ π(∆j )
and σj0 = σj .
if x ∈
6 π(∆j )

0
Proof: Since Qj ⊂ Q0j , then ρ0j (x) ≤ ρj (x) and σj (x) ≤ σj0 (x) for every x ∈ π(Q
Pn j ).
If xP
∈ π(∆j ), there exists xn ≥ 0 such that (x, xn ) ∈ ∆j . Then,
(x, xn ) = i=1 ti λij ei ,
Pn−1
where ni=1 ti = 1 and ti ≥ 0 for every 1 ≤ i ≤ n. Taking y = i=1
ti λij ei , we have that
0
0
π(y) = x, (y)n = 0 and y ∈ Qj . Hence, ρj (x) = 0.
Consider now x ∈ π(Qj )\π(∆j ). Take (x, ρ0j (x)) ∈ Q0j = conv(Qj ∪ {0}). Then,
(x, ρ0j (x)) = tq, with q ∈ Qj and 0 < t ≤ 1. Since (x, ρ0j (x)) ∈
/ ∆j , there exists 0 < t′ < 1
such that t′ (x, ρ0j (x)) lies in the nontrivial facet of ∆j and so, q ′ := t′ (x, ρ0j (x)) ∈ Qj .
Then, the line segment qq ′ is contained in Qj ; in particular, (x, ρ0j (x)) ∈ Qj . It follows
that ρj (x) ≤ ρ0j (x).
If x ∈ π(Qj ) = π(Q0j ) ⊂ Rn−1 , consider (x, σj0 (x)) ∈ Q0j . Then, (x, σj0 (x)) = tq
with q ∈ Qj and 0 ≤ t ≤ 1. If y = tq + (1 − t)λnj en ∈ Qj , then π(y) = x and so,

σj (x) ≥ (y)n = σj0 (x) + (1 − t)λnj ≥ σj0 (x).

Now, we can restate the formula for the multiplicity of the origin in Proposition 5 as
a mixed integral of convex functions:
Theorem 10 Let A = (A1 , . . . , An ) be a family of finite sets in (Z≥0 )n satisfying assumptions (H1) and (H3). Let f = (f1 , . . . , fn ) be a generic system of sparse polynomials in
10

C[x1 , . . . , xn ] supported on A. For every 1 ≤ j ≤ n, let ρ̄j be the convex function defined
in (4). Then, the origin is an isolated common zero of f and
mult0 (f ) = M In′ (ρ1 , . . . , ρn ).
Proof: By Proposition 5, it suffices to show that M Vn (A0 )− M Vn (A) = M In′ (ρ1 , . . . , ρn ).
For every 1 ≤ j ≤ n, considerPa constant νj ∈ R such
P that νj ≥ max(ρj ) ≥ max(ρj ).
For J ⊂ {1, . . . , n}, let DJ = j∈J π(∆j ) and νJ = j∈J νj . Since νJ ≥ max(⊞′j∈J ρj ),
we have that
Z


⊞′j∈J ρj dx1 . . . dxn = νJ V oln−1 (DJ ) − V oln (DJ )⊞′j∈J ρj ,νJ .
DJ

Then, by Lemma 8,
Z

⊞′j∈J ρj dx1 . . . dxn = V oln ((DJ )0,νJ ) − V oln (DJ )(⊞′j∈J ρj )|D
DJ

Now, if PJ =

P

j∈J

π(Q0j ) =

P

j∈J

J

,νJ


= V oln (PJ )⊞′



0
0
j∈J ρj ,⊞j∈J σj

Finally, by Remark 1,



= V oln ((PJ )⊞′

0
j∈J ρj ,νJ

.

) − V oln ((PJ )⊞′j∈J ρj ,νJ )



− V oln (PJ )⊞′j∈J ρj ,⊞j∈J σj .

V oln (PJ )⊞′
and

,νJ

π(Qj ), Lemma 9 implies that


V oln ((DJ )0,νJ ) − V oln (DJ )(⊞′j∈J ρj )|D



J



0
0
j∈J ρj ,⊞j∈J σj



= V oln

X
j∈J

Q0j



X 


Qj ,
V oln (PJ )⊞′j∈J ρj ,⊞j∈J σj = V oln
j∈J

and so,

Z

DJ

⊞′j∈J ρj dx1 . . . dxn = V oln

X
j∈J

X 

Qj .
Q0j − V oln
j∈J

The theorem follows from the definitions of the mixed integral and the mixed volume.

Example 2 Consider the generic sparse polynomial system
f1 = c1,20 x21 + c1,11 x1 x2 + c1,04 x42 + c1,13 x1 x32 + c1,33 x31 x32
f2 = c2,40 x41 + c2,21 x21 x2 + c2,04 x42 + c2,25 x21 x52 + c2,13 x1 x32
with supports
A1 = {(2, 0), (1, 1), (0, 4), (1, 3), (3, 3)},

A2 = {(4, 0), (2, 1), (0, 4), (2, 5), (1, 3)}.

Here, we have ∆1 = conv{(0, 0), (2, 0), (0, 4)} and ∆2 = conv{(0, 0), (4, 0), (0, 4)}.
11

conv(A2 )
conv(A1 )

π(∆1 )

π(∆2 )

To compute the multiplicity of the origin following Theorem 10, consider the convex
functions ρ1 : π(∆1 ) → R and ρ2 : π(∆2 ) → R:

ρ1 ⊞ ρ2
ρ1

ρ2

Therefore,
mult0 (f ) = M I2′ (ρ1 , ρ2 ) =

Z

0

6

ρ1 ⊞ ρ2 (x) dx −

Z

2
0

ρ1 (x) dx −

Z

4
0

ρ2 (x) dx = 7.

Remark 11 The computation of the multiplicity of the origin by means of mixed integrals
following Theorem 10 may involve smaller polytopes than its computation using mixed volumes according to Proposition 5, since it depends only on the points of the lower envelopes
of the polytopes Qj = conv(Aj ) that lie above the simplices π(∆j ) for j = 1, . . . , n.

3.2

General case

Consider now a family A = (A1 , . . . , An ) of finite sets in (Z≥0 )n satisfying conditions (H1)
and (H2). Let f = (f1 , . . . , fn ) be a system of generic sparse polynomials in C[x1 , . . . , xn ]
supported on A.
M
For M ∈ Z>0 , let ∆M := {M ei }ni=1 and, for all 1 ≤ j ≤ n, let A∆
:= Aj ∪ ∆M and
j

M ,0
∆M
∆M ,0 := (A∆M ,0 , . . . , A∆M ,0 ).
M
M
A∆
:= A∆
∪ {0}. Set A∆M := (A∆
n
1 , . . . , An ) and A
1
j
j

Proposition 12 With the previous assumptions and notation, we have that 0 is an isolated common zero of f and, for every M ≫ 0, its multiplicity is
mult0 (f ) = M Vn (A∆M ,0 ) − M Vn (A∆M ).
12

Moreover, the identity holds for every M > mult0 (f ). In particular, it suffices to take
M = M Vn (A0 ) − M Vn (A) + 1.
Proof: Conditions (H1) and (H2) imply that 0 is an isolated common zero of the generic
system f supported on A.
Take M > mult0 (f ) and consider polynomials
gj = fj +

n
X

cj,M ei xM
i

i=1

M
with support sets A∆
= Aj ∪ ∆M and generic coefficients for all 1 ≤ j ≤ n.
j
∆
∆
M
M
Since A M = (A1 , . . . , A∆
n ) fulfills the conditions (H1) and (H3) stated in Section
3.1, by Proposition 5 the multiplicity of the origin as a common isolated root of g :=
(g1 , . . . , gn ) is mult0 (g) = M Vn (A∆M ,0 ) − M Vn (A∆M ).
Let us prove that mult0 (f ) = mult0 (g). To do so, we consider the matrices Sk (f , 0) and
Sk (g, 0), for k ≥ 0, introduced in Section 2.3. Note that, since M > mult0 (f ), in order to
compute mult0 (f ), it suffices to compare the dimensions of the nullspaces of the matrices
Sk (f , 0) for 0 ≤ k ≤ M − 1. Now, for every k ≤ M − 1, α, β ∈ (Z≥0 )n , with |α| ≤ k and
|β| ≤ k − 1, and every 1 ≤ j ≤ n, we have that

(Sk (f , 0))(β,j),α =

1 ∂ |α| β
1 ∂ |α| β
(x
f
)(0)
=
(x gj )(0) = (Sk (g, 0))(β,j),α .
j
α! ∂xα
α! ∂xα

Since the dimensions of the nullspaces of Sk (f , 0) = Sk (g, 0) stabilize for k < M , then,
mult0 (f ) = mult0 (g).
The fact that we can take M = M Vn (A0 ) − M Vn (A) + 1 follows from inequality (2).

From the previous result and Theorem 10 we can express the multiplicity of the origin
as an isolated zero of a generic sparse system via mixed integrals:
Corollary 13 Let A = (A1 , . . . , An ) be a family of finite sets in (Z≥0 )n satisfying assumptions (H1) and (H2). Let f = (f1 , . . . , fn ) be a generic family of polynomials in
C[x1 , . . . , xn ] supported on A. Let M := M Vn (A0 ) − M Vn (A) + 1 and, for 1 ≤ j ≤ n,
M
let ρ∆
be the convex function that parameterizes the lower envelope of the polytope
j
∆M
conv(Aj ) and ρj ∆M its restriction defined as in (4). Then,
mult0 (f ) = M In′ (ρ1 ∆M , . . . , ρn ∆M ).
The following property enables us to deal with smaller support sets when computing
multiplicities.
Proposition 14 Let f = (f1 , . . . , fn ) be a generic system of polynomials in C[x1 , . . . , xn ]
supported on a family A = (A1 , . . . , P
An ) of finite subsets of (Z≥0 )n . Assume that 0 is an
isolated common zero of f . Let f1 = a∈A1 c1,a xa . If α, α + β ∈ A1 with β ∈ (Z≥0 )n \ {0},
then
mult0 (f ) = mult0 (f1 − c1,α+β xα+β , . . . , fn ).
13

Proof: Let h1 , . . . , hn be polynomials of the form hj = fj +
generic coefficients and M ∈ N sufficiently big such that

Pn

i=1 cj,M ei

xM
i with cj,M ei ∈ C

mult0 (f1 , . . . , fn ) = mult0 (h1 , . . . , hn ),
mult0 (f1 − c1,α+β xα+β , . . . , fn ) = mult0 (h1 − c1,α+β xα+β , . . . , hn ),
α + β 6= M ei for all 1 ≤ i ≤ n and A1 ⊂ conv({0, M e1 , . . . , M en }). The existence of M is
ensured by Proposition 12 and its proof.
To prove that mult0 (h1 , . . . , hn ) = mult0 (h1 − c1,α+β xα+β , . . . , hn ), by Proposition 5, it
suffices to show that conv(A1 ∪ {M ei }ni=1 \{α + β}) = conv(A1 ∪ {M ei }ni=1 ). This follows
from the fact that α + β ∈ conv({α, M e1 , . . . , M en }), since

α+β = 1−


X  βi
|β|αi
|β| 
α+
M ei
+
M − |α|
M
(M − |α|)M
n

i=1

is a convex linear combination of α, M e1 , . . . , M en .



As a consequence of Proposition 14 we are able to obtain a refined formula for the
multiplicity of the origin for generic polynomials supported on a family A satisfying conditions (H1) and (H2), with no need of adding extra points to the supports whenever they
intersect the coordinate axes.
Proposition 15 Let A = (A1 , . . . , An ) be a family of finite subsets of (Z≥0 )n satisfying
assumptions (H1) and (H2), and let f = (f1 , . . . , fn ) be a generic sparse polynomial system
supported on A. Let M ∈ Z, M ≥ M Vn (A0 )−M Vn (A)+1. Then, 0 is an isolated common
zero of f with multiplicity
M,0
M
M
mult0 (f ) = M Vn (AM,0
1 , . . . , An ) − M Vn (A1 , . . . , An ),

where, for every 1 ≤ j ≤ n, AM
j := Aj ∪ M ei : 1 ≤ i ≤ n, Aj ∩ {µei | µ ∈ Z≥0 } = ∅ and
AM,0
:= AM
j ∪ {0}.
j

Example 3 Consider the generic polynomial system f = (f1 , f2 , f3 ) with
f1 = c11 x1 + c12 x2 + c13 x22 + c14 x21 x2 x3
f2 = c21 x21 + c22 x31 + c23 x21 x2 + c24 x33
f3 = c31 x1 + c32 x1 x2 + c33 x23 + c34 x2 x33
with support family A = (A1 , A2 , A3 ), where
A1 = {(1, 0, 0), (0, 1, 0), (0, 2, 0), (2, 1, 1)}
A2 = {(2, 0, 0), (3, 0, 0), (2, 1, 0), (0, 0, 3)}
A3 = {(1, 0, 0), (1, 1, 0), (0, 0, 2), (0, 1, 3)}
which satisfies assumptions (H1) and (H2). Then, 0 is an isolated common root of f . In
order to compute its multiplicity according to Proposition 15, let
M := M V3 (A0 ) − M V3 (A) + 1 = 28 − 22 + 1 = 7,
14

and consider the modified support sets
A71 = {(1, 0, 0), (0, 1, 0), (0, 2, 0), (2, 1, 1), (0, 0, 7)}
A72 = {(2, 0, 0), (3, 0, 0), (2, 1, 0), (0, 0, 3), (0, 7, 0)} ,
A73 = {(1, 0, 0), (1, 1, 0), (0, 0, 2), (0, 1, 3), (0, 7, 0)}
which coincide with the supports of the polynomials in Example 1. Therefore,
7,0
7,0
7
7
7
mult0 (f ) = M V3 (A7,0
1 , A2 , A3 ) − M V3 (A1 , A2 , A3 ) = 3.

4

Multiplicity of other roots with zero coordinates

Let A = (A1 , . . . , An ) be a family of finite sets in (Z≥0 )n and f = (f1 , . . . , fn ) ⊂
C[x1 , . . . , xn ] a generic family of polynomials with support set A.
For I ⊂ {1, . . . , n}, recall that
JI = {j ∈ {1, . . . , n} | ∃a ∈ Aj : ai = 0 ∀i ∈ I}
is the set of indices of the polynomials in f that do not vanish identically under the
specialization xi = 0 for every i ∈ I. Also, for every j ∈ JI , we denote
AIj = {a ∈ Aj | ai = 0 ∀i ∈ I}.
Following [8, Section 3.2.1], the system f has isolated common zeros lying in OI :=
{x ∈ Cn | xi = 0 if and only if i ∈ I} if and only if
(A1) #I + #JI = n,
(A2) for every Ie ⊂ I, #Ie + #JIe ≥ n,
P
(A3) for every J ⊂ JI , dim( j∈J AIj ) ≥ #J.

From now on, we will consider a non-empty set I ⊂ {1, . . . , n} satisfying the conditions
above and we will study the multiplicity of the isolated common zeros of f lying in OI .

4.1

Multiplicity of affine isolated roots

The aim of this section is to compute multiplicities of the isolated zeros of f in OI in terms
of mixed volumes and mixed integrals associated to the system supports. The key result
that allows us to do this shows that these multiplicities coincide with the multiplicity of
the origin as an isolated root of an associated generic sparse system:
Theorem 16 Let A = (A1 , . . . , An ) be a family of finite sets in (Z≥0 )n and f = (f1 , . . . , fn )
a generic sparse system of polynomials in C[x1 , . . . , xn ] supported on A. Assume that
∅=
6 I ⊂ {1, . . . , n} satisfies conditions (A1), (A2) and (A3). Let ζ ∈ Cn be an isolated
zero of f with ζ ∈ OI . Then
multζ (f ) = mult0 (g),
I
for a system g := (gj )j ∈J
/ I of generic polynomials with supports Bj := πI (Aj ) for every
j∈
/ JI , where πI : Zn → Z#I is the projection onto the coordinates indexed by I.

15

For this statement to make sense, we need the following:
Lemma 17 Under the previous assumptions and notation, let B I = (BjI )j ∈J
/ I . Then,
#I
I
0 ∈ C is an isolated zero of a generic polynomial system supported on B .
Proof: It suffices to show that B I satisfies conditions (H1) and (H2) stated at the beginning of Section 3 (see [8, Proposition 6]).
By the definition of JI , it follows that 0 ∈
/ πI (Aj ) = BjI for every j ∈
/ JI .
In order to simplify notation, we will index the coordinates of Z#I by the corresponding
elements of I.
To prove that condition (H2) holds, we must show that #Ie + #JIe(B I ) ≥ #I for every
e Now, for every Ie ⊂ I, we have
/ JI | ∃b ∈ BjI : bi = 0 ∀i ∈ I}.
Ie ⊂ I, where JIe(B I ) = {j ∈
that
e = JI ∪ J e(B I ).
/ JI | ∃a ∈ Aj : ai = 0 ∀i ∈ I}
JIe(A) = JI ∪ {j ∈
I

Under assumption (A2) on I, the inequality #Ie + #JIe(A) ≥ n holds; then,
#Ie + #JIe(B I ) = #Ie + #JIe(A) − #JI ≥ n − #JI = #I,

where the last identity follows from assumption (A1).



In order to prove Theorem 16, we first introduce some notation and prove some auxiliary results.
For a polynomial g ∈ C[x1 , . . . , xn ], gI will denote the polynomial in C[(xi )i6∈I ] obtained
from g by specializing xi = 0 for every i ∈ I, and fI the associated family of polynomials
fI = ((fj )I )j∈JI .
Then, fI is the set of polynomials obtained by specializing the variables indexed by I to 0
in the polynomials in f and discarding the ones that vanish identically, and AI = (AIj )j∈JI
is the family of supports of fI .
We will use an auxiliary polynomial system defined as follows:
(
(fj )I if j ∈ JI
f (I) = (f1,I , . . . , fn,I ), where fj,I =
.
fj
if j ∈
/ JI
Note that the family of supports of these polynomials is
(
A(I) = (A1,I , . . . , An,I ), where Aj,I =

AIj
Aj

if j ∈ JI
.
if j ∈
/ JI

Lemma 18 Under the previous assumptions and notation, if ζ ∈ Cn is an isolated zero
of f lying in OI , then ζ is also an isolated zero of f (I) and multζ (f ) ≤ multζ (f (I)).
Proof: The fact that ζ is an isolated zero of f (I) follows from the facts that f (I) is a generic
system supported on A(I) vanishing at ζ, and that, for every Ie ⊂ I, JIe(A(I)) = JIe(A).
The inequality between the multiplicities is a consequence of Lemma 3.

16

We now focus on a special case of polynomial systems with the same structure as f (I),
namely, systems of n polynomials in n variables which contain r polynomials depending
only on r variables.
Proposition 19 Let h = (h1 , . . . , hn ) be a system of polynomials in C[x1 , . . . , xn ] such
that h1 , . . . , hr ∈ C[x1 , . . . , xr ]. Let ξ ∈ Cr be an isolated nondegenerate common zero
of h1 , . . . hr such that 0 ∈ Cn−r is an isolated zero of hξ := (hr+1 (ξ, xr+1 , . . . , xn ), . . . ,
hn (ξ, xr+1 , . . . , xn )). Then, ζ = (ξ, 0) ∈ Cn is an isolated zero of h satisfying:
multζ (h) = mult0 (hξ ).
Proof: Under our assumptions, it follows that ζ = (ξ, 0) is an isolated zero of the system
h: if there is an irreducible curve C passing through ζ, since ξ is an isolated common zero
of h1 , . . . , hr ∈ C[x1 , . . . , xr ], we have that C ⊂ {x1 = ξ1 , . . . , xr = ξr } and so, (ξ, 0) ∈
C ⊂ {x1 = ξ1 , . . . , xr = ξr , hr+1 (x) = 0 . . . , hn (x) = 0} = {ξ} × V (hξ ), contradicting the
fact that 0 is an isolated zero of hξ .
In order to prove the stated equality of multiplicities, we will compare the multiplicity
matrices Sk (h, ζ) and Sk (hξ , 0) for k ∈ N (see Section 2.3 for the definition of multiplicity
matrices). To this end, we will analyze the structure of Sk (h, ζ).
Recall that for the system h, for k ≥ 1, the columns of Sk (h, ζ) are indexed by α
for |α| ≤ k and its rows are indexed by (β, j) for |β| ≤ k − 1 and 1 ≤ j ≤ n; the entry
corresponding to row (β, j) and column α is
(Sk (h, ζ))(β,j),α = ∂α ((x − ζ)β hj )(ζ),
where ∂α is defined in (1).
Note that, for γ = (γ1 , . . . , γn ) ∈ (Z≥0 )n , we have
 r
Q γi  γi +βi −αi


if βi ≤ αi ≤ βi + γi ∀ 1 ≤ i ≤ r

αi −βi ξi

|α|
1 ∂
i=1
β γ
and αi = βi + γi ∀ r + 1 ≤ i ≤ n, (7)
((x − ζ) x )(ζ) =

α! ∂xα


0
otherwise.

Then, an entry of Sk (h, ζ) corresponding to a row indexed by (β, j) and a column
indexed by α is 0 whenever |β| ≥ |α|.
We will first consider the columns of Sk (h, ζ) indexed by α = (0, . . . , 0, αr+1 , . . . , αn ) 6=
0. For 1 ≤ j ≤ r, since the polynomial hj does not depend on the variables xr+1 , . . . , xn ,
we have that (Sk (h, ζ))(β,j),α = 0 for every β. For r + 1 ≤ j ≤ n and β with βi 6= 0 for
some 1 ≤ i ≤ r, we also have (Sk (h, ζ))(β,j),α = 0 since βi > αi = 0 (see equation (7)).
Finally, for r + 1 ≤ j ≤ n and β = (0, . . . , 0, βr+1 , . . . , βn ),
(Sk (h, ζ))(β,j),α =

1
∂ |α|
β
x r+1 . . . xβnn hj (ξ, xr+1 , . . . , xn )(0)
r+1
α! ∂xαr+1
. . . ∂xαnn r+1

(8)

= (Sk (hξ , 0))((βr+1 ,...,βn),j),(αr+1 ,...,αn ) .
We analyze now the remaining columns of the matrix.
Consider the submatrix of Sk (h, ζ) given by the columns indexed by α such that
(α1 , . . . , αr ) 6= 0 and |α| = k. From identity (7), we can observe that in every row
17

indexed by (β, j) for |β| = k − 1 and 1 ≤ j ≤ n, the only columns with (possibly) non-zero
coordinates are indexed by α = β +ei where {ei }ni=1 is the canonical basis of Rn ; moreover,
(Sk (h, ζ))(β,j),β+ei =

∂hj
(ζ).
∂xi
∂h

Note that, for 1 ≤ j ≤ r and r + 1 ≤ i ≤ n, we have ∂xji ≡ 0. Then, for every β with
|β| = k − 1, in
 indexed by (β, j) for 1 ≤ j ≤ r we have a copy of the Jacobian
 the rows
∂hj
in the columns indexed by β + e1 , . . . , β + er , and all other
matrix J := ∂xi (ξ)
1≤j,i≤r

entries of the matrix Sk (h, ζ) in these rows are zero. We remark that J is an invertible
matrix since ξ is a nonsingular common zero of h1 , . . . hr . Note that, for every α with
|α| = k and αi ≥ 1 for some 1 ≤ i ≤ r, there is at least one β = α − ei with |β| = k − 1;
so, all the columns indexed by α with |α| = k and (α1 , . . . , αr ) 6= 0 are involved in at least
one of the copies of J .
Therefore, by performing row operations in Sk (h, ζ) we can obtain a matrix such that
each column indexed by a vector α with |α| = k and (α1 , . . . , αr ) 6= 0 contains all zero
entries except for a unique coordinate equal to 1 in a row indexed by (β, j) for some β
with |β| = k − 1 and 1 ≤ j ≤ r, and all these 1’s lie in different rows. Moreover, these row
operations do not modify the remaining columns of Sk (h, ζ).
Then, the dimension of the kernel of Sk (h, ζ) is the same as the dimension of the kernel
of the matrix obtained by removing the columns indexed by α with (α1 , . . . , αr ) 6= 0 and
|α| = k. We repeat this procedure for s = k, k − 1, . . . , 1 (in this order) and we conclude
that the dimension of the kernel of Sk (h, ζ) is the same as the dimension of the kernel
of the submatrix obtained by removing all columns indexed by α with (α1 , . . . , αr ) 6= 0.
This submatrix consists of the first column of Sk (h, ζ), which is identically zero, and all
columns indexed by α = (0, . . . , 0, αr+1 , . . . , αn ) 6= 0. Due to our previous considerations
on the matrix formed by these columns, we have that the only rows that are not zero are
those indexed by (β, j) with r + 1 ≤ j ≤ n and β = (0, . . . , 0, βr+1 , . . . , βn ) and these are
exactly the rows of Sk (hξ , 0) (see identity (8)). Therefore,
dim(ker(Sk (h, ζ))) = dim(ker(Sk (hξ , 0))) for every k ≥ 1.
The result follows.



Now we can prove Theorem 16.
Proof: Without loss of generality, we may assume that I = {r + 1, . . . , n} for some
r ∈ {1, . . . , n} and JI = {1, . . . , r}.
We will first prove that multζ (f ) ≥ mult0 (g).
We make the change of variables
P
x1 := ri=1 c1i yi + ζ1
xr+1 := yr+1
..
..
.
Pr .
xr := i=1 cri yi + ζr
xn := yn

where (cki )1≤k,i≤r ⊂ Q are generic constants and obtain the polynomial system e
f =
e
e
e
(f1 , . . . , fn ) in C[y1 , . . . , yn ] from the system f . Note that mult0 (f ) = multζ (f ).
18

For every 1 ≤ j ≤ n, let Aej be the support of fej .
For 1 ≤ j ≤ r, since fj (x1 , . . . , xr , 0, . . . , 0) 6= 0 and has a non-constant term (since it
vanishes at (ζ1 , . . . , ζr ) ∈ (C∗ )r ), due to the genericity of the coefficients and the change
of variables, we have that the monomials y1 , . . . , yr appear with non-zero coefficients in
fej (y). On the other hand, again, for the genericity of coefficients and change of variables,
for r + 1 ≤ j ≤ n,
πI (Aej ) = πI (Aj );
(9)

moreover, taking into account that fej (0, . . . , 0, yr+1 , . . . , yn ) = fj (ζ1 , . . . , ζr , xr+1 , . . . , xn ),
we conclude that
{β ∈ (Z≥0 )n−r / (0, β) ∈ Aej )} = πI (Aj ).
(10)

Let h = (h1 , . . . , hn ) be a generic polynomial system with supports Ae = (Ae1 , . . . , Aen ).
e Let us see that Ae also satisfies condition (H2),
Note that condition (H1) holds for A.
e < n,
which implies that 0 is an isolated zero of h. For Ie ⊂ {1, . . . , n}, if #Ie + #JIe(A)
e we obtain a system in n − #Ie unknowns with
when setting yi = 0 in e
f for every i ∈ I,
e < n − #Ie equations. This system vanishes at 0 and defines a positive dimensional
#JIe(A)
variety, contradicting the fact that 0 ∈ Cn is an isolated common zero of e
f . By Lemma 3,
e
the inequality mult0 (f ) ≥ mult0 (h) holds.
Applying Proposition 14 to the polynomials in the system h, since the monomials
y1 , . . . , yr appear with non-zero coefficients in h1 , . . . , hr and, for r + 1 ≤ j ≤ n, the
supports supp(hj ) = supp(fej ) satisfy conditions
P (9) and (10), it follows that mult0 (h) =
e = (e
mult0 (e
g, g), where g
g1 , . . . , ger ) with gej = ri=1 ϑji yi + pj (yr+1 , . . . , yn ) for 1 ≤ j ≤ r,
and g = (gr+1 , . . . , gn ) with gj ∈ C[yr+1 , . . . , yn ] a generic polynomial with support πI (Aj )
for r + 1 ≤ j ≤ n.
Then, if A is the inverse of the matrix (ϑji ) and
A.(e
g1 , . . . , ger )t = (y1 + q1 (yr+1 , . . . , yn ), . . . , yr + qr (yr+1 , . . . , yn ))t ,

the following is an isomorphism:

Q[y1 , . . . , yn ]/(e
g, g) → Q[yr+1 , . . . , yn ]/(g)
7→ −qi for all 1 ≤ i ≤ r
7
→
yi for all r + 1 ≤ i ≤ n

yi
yi
and hence mult0 (e
g, g) = mult0 (g).
Therefore,

multζ (f ) = mult0 (e
f ) ≥ mult0 (h) = mult0 (e
g, g) = mult0 (g).

To prove the other inequality, note that, by Lemma 18, we have that
multζ (f ) ≤ multζ (f (I)).

Then, applying Proposition 19 to the system f (I) and ξ = (ζ1 , . . . , ζr ), we deduce that
multζ (f (I)) = mult0 (f (I)ξ ).
By the genericity of the coefficients of f and the triangular structure of f (I), the system
I , . . . , BI .
f (I)ξ turns to be a generic system supported on Br+1
n
We conclude that multζ (f ) ≤ mult0 (g).

19

Taking into account that the results in Section 3 enable us to express the multiplicity
of the origin as an isolated zero of a generic sparse system in terms of mixed volumes and
mixed integrals, we can now state a similar result regarding the multiplicity of any affine
isolated zero of a generic sparse system of n equations in n unknowns.
Theorem 20 Let A = (A1 , . . . , An ) be a family of finite sets in (Z≥0 )n and f = (f1 , . . . , fn )
be a generic sparse system of polynomials in C[x1 , . . . , xn ] supported on A.
Let I ⊂ {1, . . . , n} satisfying conditions (A1), (A2) and (A3). For j ∈
/ JI , let BjI =
#I is the projection to the coordinates indexed by I. Let
πI (Aj ), where πI : Zn → Z


I
MI := M V#I (Bj ∪ {0})j6∈JI − M V#I (BjI )j6∈JI + 1.
Then, for every isolated zero ζ ∈ Cn of f such that ζi = 0 if and only if i ∈ I, we have
#I
I
multζ (f ) = M V#I ((BjI ∪ {0, MI ei }#I
/ I ) − M V#I ((Bj ∪ {MI ei }i=1 )j ∈J
/ I ).
i=1 )j ∈J

Moreover, if (ρj )j6∈JI are the convex functions that parameterize the lower envelopes of the
polytopes conv(BjI ∪ {MI ei }#I
/ I are their restrictions as defined in (4), then
i=1 ) and (ρj )j ∈J
′ ((ρ )
multζ (f ) = M I#I
).
/ I
j j ∈J
Note that the previous formula for multiplicities can be refined applying Proposition
15 instead of Proposition 12.

4.2

Examples

The following examples illustrate the result in the previous section.
Example 4 Consider the generic polynomial system

 c11 x21 + c12 x21 x22 + c13 x1 x3 + c14 x1 x22 x3 + c15 x43 + c16 x22 x43 = 0
c x4 + c22 x41 x22 + c23 x21 x3 + c24 x21 x22 x3 + c25 x43 + c26 x22 x43 = 0
 21 1
c31 x1 + c32 x1 x22 + c33 + c34 x22 + c35 x3 + c36 x22 x3 = 0

taken from [7, Example 3]. There is a unique nonempty set I = {1, 3} satisfying conditions
(A1), (A2) and (A3), which leads to two isolated solutions with x1 = 0, x2 6= 0 and
x3 = 0. Since JI = {3}, Theorem 16 tell us that the multiplicity of each of these solutions
equals the multiplicity of (0, 0) as an isolated root of a generic sparse system supported on
B1I = {(2, 0), (1, 1), (0, 4)} and B2I = {(4, 0), (2, 1), (0, 4)}, namely a system of the type

a1 x21 + b1 x1 x3 + c1 x43 = 0
a2 x41 + b2 x21 x3 + c3 x43 = 0
This multiplicity can be computed, by Proposition 5, as M V2 (B1I ∪{(0, 0)}, B2I ∪{(0, 0)})−
M V2 (B1I , B2I ) = 7 or, alternatively, by Theorem 10, as M I2′ (ρ1 , ρ2 ) = 7, where ρ1 and ρ2
are the functions whose graphs are given in Example 2.
Example 5 Consider the generic polynomial system

a11 x1 + a12 x1 x2 = 0



a21 x22 + a22 x21 x42 + a23 x31 = 0
 a31 x3 + a32 x1 x3 + a33 x23 x24 + a34 x33 x4 = 0


a41 x34 + a42 x32 x34 + a43 x23 x34 + a44 x54 + a45 x23 x54 = 0
20

Using [8, Proposition 5] we can check that all zeros of the system are isolated. Moreover,
all the subsets I ⊂ {1, 2, 3, 4} satisfying conditions (A1), (A2) and (A3) are
I1 = ∅, I2 = {3}, I3 = {1, 2}, I4 = {3, 4}, I5 = {1, 2, 3} and I6 = {1, 2, 3, 4}.
By Bernstein’s theorem, the system has 24 different simple zeros with all non-zero coordinates (associated to I1 ) and, by Theorem 20, we can see that there are
• 6 simple zeros associated to I2 ,
• 8 zeros with multiplicity 2 associated to I3 ,
• 3 zeros with multiplicity 3 associated to I4 ,
• 2 zeros with multiplicity 2 associated to I5 ,
and that the origin is an isolated zero of multiplicity 6.
That is, the system has a total of 65 (isolated) zeros counting multiplicities. Note that,
in this case, SM4 (A) = 65 is smaller than M V4 (A ∪ {0}) = 85.

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