Elimination for Generic Sparse Polynomial Systems
Discrete Comput Geom
Elimination for Generic Sparse Polynomial Systems
María Isabel Herrero 0
Gabriela Jeronimo 0
Juan Sabia 0
0 M.I. Herrero
We present a new probabilistic symbolic algorithm that, given a variety defined in an ndimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an dimensional coordinate affine space with < n. The complexity of the algorithm depends polynomially on some combinatorial invariants associated to the supports. Let A = (A1, . . . , Ar ) be a family of finite subsets of (Z≥0)n and f = (f1, . . . , fr ) a system of polynomials in Q[X1, . . . , Xn] supported on A. If V (f) ⊂ Cn denotes the affine variety of the common zeros of the polynomials in f and, for a given < n,
Projection of algebraic varieties; Sparse polynomial systems; Algorithms and complexity

1 Introduction
π : Cn → C is the projection π(x1, . . . , xn) = (x1, . . . , x ), we consider the
problem of computing algorithmically a description of the Zariski closure π(V (f)) ⊂ C
within a complexity depending on combinatorial invariants associated to the input
supports.
The computation of (Zariski closures of) linear projections of varieties is the
basic task in elimination theory. A more general formulation of this problem is
algorithmic quantifier elimination over algebraically closed fields (see, for instance,
[
9, 18, 23, 36, 44
] for algorithms with complexities depending on the number and
degrees of the polynomials and the number of variables involved). Particular instances
of the computation of Zariski closures of projections are the computation of Chow
forms (see, for instance, [
7, 21, 29
]), classical resultants (see [
13
] and the references
therein) and sparse resultants (see, for example, [
8, 12, 28, 43
]).
The foundations of the study of sparse polynomial systems can be traced back to
[
4, 31
] and [32], which prove that the number of isolated roots in (C∗)n of a system
of n polynomial equations in n unknowns is bounded by the mixed volume of the
family of their supports.
This result led to the construction of algorithms for the computation of these
isolated solutions, particularly designed to deal with sparse systems (see, for instance,
[
26, 30, 38, 46
]), which run faster than the known general procedures solving this
task. Later on, upper bounds for the number of isolated affine solutions of sparse
systems were obtained and new efficient algorithms to compute them were designed (see,
for instance, [
17, 20, 24, 27, 34, 37, 39
]). More recently, positive dimensional
components of affine varieties defined by sparse systems were also considered: in [45],
certificates for the existence of 1dimensional components were given and, in [
1
]
and [
2
], under certain assumptions on the equations, algorithmic methods to describe
Puiseux series expansions of curves and arbitrary positive dimensional components,
respectively, were presented. Also, in [
25
], an upper bound in terms of mixed
volumes for the degree of the affine variety defined by a sparse polynomial system of
n equations in n unknowns was proved and algorithms for characterizing the
equidimensional decomposition of affine varieties defined by sparse systems were designed.
The good performance of the most efficient algorithms dealing with sparse systems
relies on the use of polyhedral deformations introduced in [
26
], because these
deformations essentially preserve the monomial structure of the system involved.
In this paper, we present a probabilistic symbolic algorithm which computes the
Zariski closure π(V (f)) ⊂ C for a sparse system f with fixed supports and generic
coefficients. We use the decomposition of the variety V (f) into equidimensional
subvarieties contained in coordinate subspaces proved in [
25
] to reduce the problem to
the case of a variety such that each of its components intersects the torus. In this case
we compute a geometric resolution of the variety with respect to a suitably chosen
set of free variables and, from this resolution, we show how to obtain a geometric
resolution of the Zariski closure of the required projection. The complexity of our
algorithm is polynomial in some combinatorial invariants associated to the supports
of the input polynomials. Our main result is the following.
Theorem 1 Let f = (f1, . . . , fr ) be a system of polynomials in Q[X1, . . . , Xn] with
generic coefficients supported on a family A = (A1, . . . , Ar ) of finite subsets of
(Z≥0)n such that dim( j∈J Aj ) ≥ #J for every J ⊂ {1, . . . , r}, and let V ∗(f) ⊂ Cn
be the Zariski closure of
x ∈
C∗ n  fj (x) = 0 for every 1 ≤ j ≤ r .
There is a probabilistic algorithm that computes a geometric resolution of π(V ∗(f)),
where π : Cn → C is the projection to the first coordinates, within
O rD n2N log(d) D2 + E + D5
operations in Q up to logarithmic factors, where
D = MV A, Δ(n−r) ,
E =
d := max deg(fj )
j
and
Here, MV denotes the mixed volume, Δ is the vertex set of the standard simplex in
Rn and the superscript indicates the number of repetitions.
The stated complexity is due to a polyhedral deformationbased algorithm to solve
generic sparse zerodimensional systems [
30
], an algorithmic Newton–Hensel lifting
[
40
] and codification of multivariate polynomials as straightline programs [
6
]. As
in [
30
], we do not take into account the cost of mixedvolume related computations
(see Sect. 2.2 for more details on this point). Furthermore, in the complexity of
Theorem 1 we have ignored terms depending on the size of certain combinatorial objects
associated to polyhedral deformations involved; for a more precise complexity
estimate, see Theorem 10.
The paper is organized as follows. In Sect. 2, the notation and the basic
theoretical and algorithmic notions used throughout are introduced. Section 3 is devoted to
proving the main theoretical results on which our algorithms rely. Finally, Sect. 4
contains the descriptions, proof of correctness, and complexity estimates of our
algorithms, and examples illustrating how they work.
2 Preliminaries
2.1 Basic Definitions and Notation
Let k be a field of characteristic zero and k be an algebraic closure of k. Given
polynomials f1, . . . , fr ∈ k[X1, . . . , Xn], we write V (f) = V (f1, . . . , fr ) for the affine
variety which is the set of the common zeros of f = (f1, . . . , fr ) in kn, and
V ∗(f) = V ∗(f1, . . . , fr ) := V (f1, . . . , fr ) ∩ k∗ n
for the union of the irreducible components of V (f1, . . . , fr ) containing points with
all their coordinates in k∗ := k − {0}.
For a variety V ⊂ kn definable over k, we denote k[V ] = k[X1, . . . , Xn]/I (V ) its
coordinate ring (where I (V ) ⊂ k[X1, . . . , Xn] is the ideal of the polynomials
vanishing identically on V ). If V is irreducible, we write k(V ) for the fraction field of k[V ].
To describe zerodimensional affine varieties we use the notion of a geometric
reskolnubtieona (zseereo,fdoirminesntsainocnea,l[v2a2r]ieatnyddthefienreefderbeyncpeoslythneormeiina)l:sLinetkV[X=1,{.ξ.(.1,),X. n..]., ξG(iDv)e}n⊂a
linear form λ = λ1X1 + · · · + λnXn in k[X1, . . . , Xn] such that λ(ξ (i)) = λ(ξ (j)) if
i = j , the following polynomials completely characterize V :
• the minimal polynomial qλ = 1≤i≤D(Y − λ(ξ (i))) ∈ k[Y ] of λ over the variety
V (where Y is a new variable),
• polynomials v1, . . . , vn ∈ k[Y ] with deg(vj ) < D for every 1 ≤ j ≤ n satisfying
V = {(v1(η), . . . , vn(η)) ∈ Cn  η ∈ k, qλ(η) = 0}.
The family of univariate polynomials (qλ, v1, . . . , vn) ∈ k[Y ]n+1 is called the
geometric resolution of V (or a geometric resolution of k[V ]) associated with the linear
form λ.
The notion of geometric resolution can be extended to any equidimensional
variety: Let V ⊂ kn be an equidimensional variety of dimension t defined by
polynomials in k[X1, . . . , Xn]. Assume that, for each irreducible component W of V , the
identity I (W ) ∩ k[X1, . . . , Xt ] = {0} holds. By considering k(X1, . . . , Xt ) ⊗ k[V ],
we are in a zerodimensional situation, and we call a geometric resolution of
V with free variables X1, . . . , Xt to a geometric resolution (qλ, vt+1, . . . , vn) ∈
k(X1, . . . , Xt )[Y ]n−t+1 of k(X1, . . . , Xt ) ⊗ k[V ] associated to a linear form λ ∈
k[Xt+1, . . . , Xn]. If qˆλ ∈ k[X1, . . . , Xt , Y ] is obtained from qλ by clearing
denominators, a geometric resolution of V gives a birational map between the hypersurface
{(x1, . . . , xt , y) ∈ kn−t+1  qˆλ(x1, . . . , xt , y) = 0} and V .
When dealing with varieties defined by sparse polynomial systems, an important
combinatorial invariant associated to the system is the mixed volume of their
supports. For a family A = (A1, . . . , An) of n finite subsets of (Z≥0)n, MVn(A) (or
MV(A) if n is clear from the context) denotes the ndimensional mixed volume of
the convex hulls in Rn of A1, . . . , An (see, for instance, [10, Chap. 7] for the
definition and basic properties). In this context, we write Δ for the vertex set {0, e1, . . . , en}
of the standard simplex of Rn, which is the support of an affine linear form with
nonzero coefficients, and Δ(t) for the family of t copies of Δ.
2.2 Algorithms and Codification
Although we work with polynomials, our algorithms only deal with elements in a
base field k. The notion of complexity of an algorithm we consider is the number of
operations and comparisons in k it performs. We will encode multivariate
polynomials in different ways:
• in sparse form, that is, by means of the list of pairs (a, ca) where a runs over a
fixed set including the exponents of the monomials appearing in the polynomial
with nonzero coefficients and ca is the corresponding coefficient,
• in the standard dense form, which encodes a polynomial of degree bounded by d
as the vector of the coefficients of all the monomials of degree at most d including
zeros (we use this encoding only for univariate polynomials),
• in the straightline program (slp for short) encoding. A straightline program is an
algorithm without branchings which allows the evaluation of the polynomial at a
generic value (for a precise definition and properties of slp’s, see [
6
]).
In our complexity estimates, we use the usual O notation: for f, g : Z≥0 → R,
f (d ) = O(g(d )) if f (d ) ≤ cg(d ) for a positive constant c. We also use the
notation M (d ) = d log2(d ) log(log(d )), where log denotes logarithm to base 2. We
recall that multipoint evaluation and interpolation of univariate polynomials of
degree d with coefficients in a commutative ring R of characteristic 0 can be performed
with O(M (d )) operations and that multiplication and division with remainder of such
polynomials can be done with O(M (d )/ log(d )) arithmetic operations in R. We write
Ω for the exponent (Ω < 4) in the complexity O(d Ω ) of operations (determinant and
adjoint computations) on d × d matrices with entries in a commutative ring R [
3
]. We
use Padé approximation in order to compute the dense representation of the
numerap
tor and denominator of a rational function f = q ∈ k(Y ) with max{deg p, deg q} ≤ d
from its Taylor series expansion up to order 2d , which we do by means of subresultant
computations within O(d Ω+1) arithmetic operations in k (see [48, Corollaries 5.21
and 6.48]).
Our algorithms are probabilistic in the sense that they make random choices of
points (which we consider costfree in our complexity estimates) which lead to a
correct computation provided the points lie outside certain proper Zariski closed sets of
suitable affine spaces. Using the Schwartz–Zippel Lemma [
41, 49
], the error
probability of our algorithms can be controlled by making these random choices within
sufficiently large sets of integer numbers whose size depend on the degrees of the
polynomials defining the previously mentioned Zariski closed sets.
We use two previous algorithms as subroutines:
• a probabilistic algorithm that, given n generic sparse polynomials in n variables
with coefficients in k, computes a geometric resolution of the set of their common
zeros in kn (see [30, Sect. 5]),
• a Newton–Henselbased procedure that, given a system f of n polynomials in n
variables and t parameters, a specialization point ξ ∈ Qt for the parameters and
a geometric resolution of the set of simple common zeros of f(ξ , ·), computes an
approximation up to a given precision of the geometric resolution of the
components of V (f) where the Jacobian determinant of the system with respect to its n
variables does not vanish identically (see [40, Sect. 4.2]).
The algorithm in [
30
] assumes that the mixed cells in a fine mixed subdivision
of the family of the input supports are given (see [
26
] for the definition of a fine
mixed subdivision), that is to say, the computation of mixed cells is considered as a
preprocessing. In this paper, our algorithms also require deciding whether a mixed
volume is zero or not, and computing mixed volumes.
While the nonvanishing of a mixed volume can be decided algorithmically in
polynomial time, the problem of computing mixed volumes is known to be #Phard
(see [
14, 16
]). The mixed cells in a fine mixed subdivision of a family of finite sets
of Zn (and, therefore, the mixed volume) can be obtained algorithmically by means
of lifting (see [26]) and linear programmingbased procedures. In [
16
], an algorithm
following this approach is presented with a worstcase complexity single exponential
in n. Successive algorithms that run faster according to numerical results can be found
in [
33
] and [
19
]. A dynamic approach which does not use a random lifting function
is given in [
47
]. The dynamic enumeration procedure from [
35
], which proved to be
efficient even for large systems, seems to be the fastest known up until now. However,
there are no explicit complexity upper bounds for these more efficient procedures,
neither depending on the input nor the output size (namely, the mixed volume).
As in [
30
], we consider the computation of mixed cells as a preprocessing and do
not include its cost in our complexity estimates. Note that if our algorithm needs to
be applied to several systems with the same supports, this preprocessing only has to
be carried out once.
(1)
3 Theoretical Results
Let n, r be positive integers and A = (A1, . . . , Ar ) a family of r finite subsets of
(Z≥0)n. Consider r polynomials in n variables X = (X1, . . . , Xn) supported on A
with indeterminate coefficients: for every 1 ≤ j ≤ r , let
fj (Cj , X) =
Cj,a Xa,
a∈Aj
where Cj := (Cj,a )a∈Aj is a set of Nj := #Aj new indeterminates. Let f :=
(f1, . . . , fr ) and K := Q(C1, . . . , Cr ).
3.1 Reduction to the Toric Case
Consider the family f = (f1, . . . , fr ) of polynomials in K[X1, . . . , Xn] supported on
A with indeterminate coefficients introduced in (1). Since f is a generic system
supported on A, the equidimensional decomposition of the affine variety V (f) ⊂ Kn
depends only on the combinatorial structure of A (see [
25
]). Moreover, its
equidimensional components can be defined from certain smaller polynomial families fI in
fewer variables associated to f. This decomposition, which we explain below, enables
us to reduce the problem of computing a projection of the affine variety V (f) to the
computation of finitely many projections of varieties having a nonempty intersection
with the torus.
For I ⊂ {1, . . . , n}, given a polynomial f ∈ K[X1, . . . , Xn], we write fI to denote
the polynomial in K[(Xi )i∈/I ] obtained from f by specializing Xi = 0 for every i ∈ I .
We define JI = {j ∈ {1, . . . , r}  ∃a ∈ Aj : ai = 0 ∀ i ∈ I } and fI = ((fj )I )j∈JI (note
that JI is the set of indices of those polynomials fj that do not vanish identically
when specializing Xi = 0 for every i ∈ I ). Let
Γ = I ⊂ {1, . . . , n}  ∀J ⊂ JI , dim
I
Aj ≥ #J ; ∀I ⊂ I, #JI + #I ≥ #JI + #I ,
j∈J
where, for every 1 ≤ j ≤ r , AIj ⊂ (Z≥0)n−#I denotes the support of (fj )I . Finally,
let ϕI : Kn−#I → Kn be the map that inserts zeros in the coordinates indexed by I .
Using the previous notation, according to [25, Theorem 7], we have
V (f) =
ϕI V ∗(fI ) .
I ∈Γ
Let π : K → K
{1, . . . , n}, let I =
n
be the projection π(x1, . . . , xn) = (x1, . . . , x ). For I ⊂
− #(I ∩ {1, . . . , }), π I : Kn−#I → K I the projection to the
first coordinates and ϕI ∩{1,..., } : K I → K the map that inserts zeros in the
coordinates indexed by I ∩ {1, . . . , }. As π(ϕI (V ∗(fI ))) = ϕI ∩{1,..., }(π I (V ∗(fI ))) holds
for every I , we conclude that
π V (f) =
ϕI ∩{1,..., } π I V ∗(fI ) .
I ∈Γ
Therefore, the computation of π(V (f)) amounts to obtaining the Zariski closures
of the projections by π I of the affine varieties V ∗(fI ) = {x ∈ (K∗)n−#I  fI (x) = 0}.
For this reason, in the sequel we will deal with this problem.
3.2 The Vanishing Ideal of the Projection
We consider a generic polynomial system f := (f1, . . . , fr ) supported on A :=
(A1, . . . , Ar ) with indeterminate coefficients Cj := (Cj,a )a∈Aj for every 1 ≤
j ≤ r as in Eq. (1). We assume that r ≤ n and that, for every J ⊂ {1, . . . , r},
dim( j∈J Aj ) ≥ #J (or, equivalently, MV(A, Δ(n−r)) > 0) so that V ∗(f) is not
empty.
Our aim is to obtain a geometric resolution of the Zariski closure π(V ∗(f)), where
n
π : K → K ( ≤ n) is the projection defined by π(x1, . . . , xn) = (x1, . . . , x ). We
begin by proving some basic results on the vanishing ideals of the varieties V ∗(f) and
π(V ∗(f)).
Lemma 2 Under the previous assumptions, (f1, . . . , fr ) : (X1 · · · Xn)∞ is a prime
ideal in K[X1, . . . , Xn] of dimension n − r . Moreover, I (V ∗(f)) = (f1, . . . , fr ) :
(X1 · · · Xn)∞.
Proof For every 1 ≤ j ≤ r , fix aj0 ∈ Aj and let Aj := Aj − {aj0} and Cj :=
Cj − {Cj,aj0 }. We denote C := (C1, . . . , Cr ) and C := (C1, . . . , Cr ). Consider the
ring morphism ψ : Q[C, X] → Q[C , X]X1···Xn (where Q[C , X]X1···Xn is the
localization of Q[C , X] at the multiplicative set {(X1 · · · Xn)m  m ∈ Z≥0}) defined by
ψ (Xi ) = Xi for 1 ≤ i ≤ n, ψ (Cj,a) = Cj,a for 1 ≤ j ≤ r , a ∈ Aj , and ψ (Cj,aj0 ) =
−X−aj0 ( a∈Aj Cj,aXa) for 1 ≤ j ≤ r .
We claim that ker(ψ ) = (f1, . . . , fr ) : (X1 · · · Xn)∞. It is clear that for a
polynomial g ∈ (f1, . . . , fr ) : (X1 · · · Xn)∞, ψ (g) = 0. Assume now that for a
polynomial g ∈ Q[C, X] we have ψ (g) = 0. Let C0 := (C1,a10 , . . . , Cr,ar0 ) and let
C0 := (C10, . . . , Cr0) be new variables. By Taylor expansion, g(C0, C , X) =
g(C0, C , X)+ 1≤j≤r (Cj0 −Cj,aj0 )·Gj for certain polynomials Gj ∈ Q[C0, C, X].
Specializing Cj0 = −X−aj0 (
ψ (g) = g(C0, C , X) −
a∈Aj Cj,aXa) for every 1 ≤ j ≤ r , it follows that
1≤j≤r X−aj0 fj · Gj with Gj ∈ Q[C, X]X1···Xn ; therefore,
g(C, X) = 1≤j≤r X−aj0 fj · Gj . Multiplying by (X1 · · · Xn)m for a sufficiently
large m, we conclude that (X1 · · · Xn)mg(C, X) ∈ (f1, . . . , fr ).
Then, (f1, . . . , fr ) : (X1 · · · Xn)∞ is a prime ideal of Q[C, X]. The first statement
of the lemma follows by localizing at Q[C] − {0}.
In order to prove the second part of the lemma, consider first a polynomial
g ∈ (f1, . . . , fr ) : (X1 · · · Xn)∞, and let m ∈ Z≥0 be such that (X1 · · · Xn)mg ∈
(f1, . . . , fr ). Then, (X1 · · · Xn)mg vanishes over V (f) and so, g vanishes over
V (f) ∩ (K∗)n; therefore, g ∈ I (V ∗(f)). Conversely, if g ∈ I (V ∗(f)), then it
vanishes over each irreducible component of V (f) intersecting (K∗)n properly. Then,
(X1 · · · Xn)g vanishes over V (f1, . . . , fr ), which implies that there exists m ∈ Z≥0
with (X1 · · · Xn)mgm ∈ (f1, . . . , fr ). Therefore, gm ∈ (f1, . . . , fr ) : (X1 · · · Xn)∞
and, since this is a prime ideal, it follows that g ∈ (f1, . . . , fr ) : (X1 · · · Xn)∞.
Corollary 3 The affine variety V ∗(f) ⊂ Kn is an irreducible K variety of dimension
n − r .
Taking into account that for any K variety V ⊂ Kn, the identity I (π(V )) =
I (V ) ∩ K[X1, . . . , X ] holds, Lemma 2 also enables us to characterize the
vanishing ideal of the projection we want to compute:
Corollary 4 With the previous assumptions and notation,
I π V ∗(f)
= (f1, . . . , fr ) : (X1 · · · Xn)∞
∩ K[X1, . . . , X ].
Let t := dim(π(V ∗(f))). Without loss of generality, by renaming variables,
we may assume that {X1, . . . , Xt } ⊂ {X1, . . . , X } is a transcendence basis of
K(π(V ∗(f))) over K and {Xt+r+1, . . . , Xn} are such that {X1, . . . , Xt , Xt+r+1,
. . . , Xn} is a transcendence basis of K(V ∗(f)) over K . The following proposition
generically allows us to deal with projections with 0dimensional generic fibers.
Proposition 5 There is a Zariski dense open set O ⊆ Kn−t−r such that, for every
b ∈ Kn−t−r ∩ O, the identity I (π(V ∗(f))) = ((f1(X1, . . . , Xt+r , b), . . . , fr (X1, . . . ,
Xt+r , b)) : (X1 · · · Xt+r )∞) ∩ K[X1, . . . , X ] holds.
Proof Let X := (X1, . . . , Xt+r ). From Corollary 4, it is clear that I (π(V ∗(f))) ⊂
((f1(X, b), . . . , fr (X, b)) : (X1 · · · Xt+r )∞) ∩ K[X1, . . . , X ] for every b = (bt+r+1,
. . . , bn) such that bi = 0 for every t + r + 1 ≤ i ≤ n.
For the converse inclusion, first note that, for every 1 ≤ j ≤ r ,
K[X1, . . . , Xn]/(f1, . . . , fj ) : (X1 · · · Xn)∞
K[X1, . . . , Xn](X1···Xn)/(f1, . . . , fj )
K[Y, X1, . . . , Xn]/(Y X1 · · · Xn − 1, f1, . . . , fj ).
As in the first part of the proof of Lemma 2, it follows that (f1, . . . , fj ) : (X1 · · · Xn)∞
is a prime ideal of dimension n − j for every 1 ≤ j ≤ r ; therefore, Y X1 · · · Xn − 1,
f1, . . . , fr is a reduced regular sequence in K[Y, X1, . . . , Xn]. Moreover, the set
{X1, . . . , Xt , Xt+r+1, . . . , Xn} is algebraically independent modulo (Y X1 · · · Xn − 1,
f1, . . . , fr ). Then, by [11, Corollary 17 and Theorem 19], there exists a K definable
Zariski dense open set O ⊂ Kn−t−r containing {xi = 0 ∀ 1 ≤ i ≤ t + r} such
that for every b ∈ O ∩ Kn−t−r , cb Y X1 · · · Xt+r − 1, f1(X, b), . . . , fr (X, b) (where
cb := bt+r+1 · · · bn) is a reduced regular sequence in K[X] and {X1, . . . , Xt } is
algebraically independent modulo each of the associated primes of the ideal this regular
sequence generates. By noticing that
K[X]/ f1(X, b), . . . , fr (X, b) : (X1 · · · Xt+r )∞
K[X]/ cb Y X1 · · · Xt+r − 1, f1(X, b), . . . , fr (X, b) ,
we conclude that (f1(X, b), . . . , fr (X, b)) : (X1 · · · Xt+r )∞ is a radical
equidimensional ideal of dimension t and {X1, . . . , Xt } is algebraically independent modulo
each of its associated primes. Then, the same holds for the ideal ((f1(X, b), . . . ,
fr (X, b)) : (X1 · · · Xt+r )∞)∩K[X1, . . . , X ]. As this ideal includes the t dimensional
prime ideal I (π(V ∗(f))), the equality in the statement of the proposition holds.
3.3 Free Variables
The following result gives a combinatorial condition for the algebraic independence
of a subset of variables modulo the vanishing ideal of V ∗(f) that we will use in our
algorithm to compute a suitable transcendence basis of the rational fraction field of
this variety.
Lemma 6 Let f1, . . . , fr ∈ K[X1, . . . , Xn] be sparse polynomials supported on a
family A = (A1, . . . , Ar ) of finite subsets of (Z≥0)n as introduced in (1), and let
I = (f1, . . . , fr ) : (X1 · · · Xn)∞ ⊂ K[X1, . . . , Xn]. Let {e1, . . . , en} be the canonical
basis of Qn.
Then, the set {Xi1 , . . . , Xik } ⊂ {X1, . . . , Xn} for 1 ≤ i1 < · · · < ik ≤ n is
algebraically independent modulo I if and only if MV(A1, . . . , Ar , {0, ei1 }, . . . , {0, eik },
Δ(n−r−k)) > 0.
Proof First, assume that MV(A1, . . . , Ar , {0, ei1 }, . . . , {0, eik }, Δ(n−r−k)) > 0. Let Q
be a nonzero polynomial in the coefficients of a system supported on (A, {0, ei1 },
. . . , {0, eik }, Δ(n−r−k)) such that for each coefficient vector (c, η) = (c1, . . . , cr , ηr+1,
. . . , ηn) with coordinates in Q∗ and Q(c, η) = 0, the corresponding sparse system has
as many isolated solutions in (C∗)n as the mixed volume.
Let p ∈ I ∩ K[Xi1 , . . . , Xik ]. Without loss of generality, we may assume that p ∈
Q[C, Xi1 , . . . , Xik ]. Then there exist m ∈ Z≥0 and a nonzero polynomial p0 ∈ Q[C]
such that
p0(C)(X1 · · · Xn)mp(C, Xi1 , . . . , Xik ) = g1f1 + · · · + gr fr
(2)
with g1, . . . , gr ∈ Q[C, X1, . . . , Xn].
For each (c, η) with coordinates in Q∗ such that p0(c)Q(c, η) = 0, considering a
solution ξ ∈ (C∗)n of the system
f1(c1, X) = 0, . . . ,
fr (cr , X) = 0,
ηr+1,1 + ηr+1,2Xi1 = 0, . . . ,
ηr+k,1 + ηr+k,2Xik = 0,
ηr+k+1,0 +
ηr+k+1,i Xi = 0, . . . ,
ηn,0 +
ηn,i Xi = 0
and specializing identity (2) in (c, η, ξ ) we obtain p(c, −ηr+1,1/ηr+1,2, . . . , −ηr+k,1/
ηr+k,2) = 0. We conclude that p ≡ 0.
Assume now that {Xi1 , . . . , Xik } is algebraically independent modulo I =
I (V ∗(f)). Let l1, . . . , ln−r−k be linear forms in the variables X1, . . . , Xn with
coefficients in K∗ such that {Xi1 , . . . , Xik , l1, . . . , ln−r−k} is a transcendence basis
of K(V ∗(f)). Since V ∗(f) ∩ (K∗)n is a dense open subset of V ∗(f), for a generic
(ζ1, . . . , ζn−r ) ∈ (K∗)n−r , the set
V ∗(f) ∩ {xij = ζj ∀ 1 ≤ j ≤ k} ∩ lj (x) = ζk+j ∀ 1 ≤ j ≤ n − r − k
is not empty and consists of finitely many points in (K∗)n. These points are the
common solutions in (K∗)n of the system
f1(X) = 0, . . . ,
fr (X) = 0,
Xi1 − ζ1 = 0, . . . ,
Xik − ζk = 0,
l1(X) − ζk+1 = 0, . . . , ln−r−k(X) − ζn−r = 0,
which is supported on (A1, . . . , Ar , {0, ei1 }, . . . , {0, eik }, Δ(n−r−k)). By Bernstein’s
Theorem, we conclude that MV(A1, . . . , Ar , {0, ei1 }, . . . , {0, eik }, Δ(n−r−k)) > 0.
Remark 7 For 1 ≤ i1 < · · · < ik ≤ n, we have
MVn A1, . . . , Ar , {0, ei1 }, . . . , {0, eik }, Δ(n−r−k)
= MVn−k
(A1), . . . , (Ar ), (Δ)(n−r−k)
(see [42, Lemma 6]), where : Rn → Rn−k is the projection to the coordinates
indexed by {1, . . . , n} − {i1, . . . , ik}. This implies that, in order to determine whether a
set of k variables is algebraically independent or not, it suffices to determine whether
an (n − k)dimensional mixed volume is zero or not.
4 Algorithms
In this section we will present an algorithm to compute the Zariski closure of the
projection of V ∗(f) to the first coordinates, where f is a generic polynomial system
with given supports. First, we describe some subroutines.
4.1 Subroutines
The first subroutine we will use, which follows from Lemma 6, computes a
transcendence basis of K(V ∗(f)) containing a transcendence basis of K(π(V ∗(f))).
Algorithm TransBasis
INPUT: A family A = (A1, . . . , Ar ) of finite subsets of (Z≥0)n such that
dim( j∈J Aj ) ≥ #J for every J ⊂ {1, . . . , r}.
1. TB := ∅
2. k := 1
3. while #TB < n − r do
(a) If MV(A1, . . . , Ar , ({0, eij })ij ∈TB, {0, ek}, Δ(n−r−#TB−1)) > 0,
TB := TB ∪ {k}.
(b) k := k + 1
OUTPUT: The set TB = {i1, . . . , in−r } with i1 < · · · < in−r such that {Xi1 , . . . , Xin−r }
is a transcendence basis of K(V ∗(f)) over K and, for each 1 ≤ j ≤ n − r ,
{Xi1 , . . . , Xij } is a maximal algebraically independent subset of {X1, . . . , Xij }.
Note that the above algorithm requires to decide whether the mixed volume of a
family of finite sets is nonzero for at most n families. Following [14, Theorem 8],
there is a polynomial time algorithm to achieve this task based on the matroid
intersection algorithm from [
15
]. Therefore, Algorithm TransBasis runs in polynomial
time.
Without loss of generality, by renaming variables, we may assume that the
transcendence basis of K(V ∗(f)) obtained by applying algorithm TransBasis is
{X1, . . . , Xt , Xt+r+1, . . . , Xn} with t ≤ and ≤ t + r . Then, {X1, . . . , Xt } is a
transcendence basis of K(π(V ∗(f))). Let fb be the polynomial system obtained by
evaluating Xt+r+1, . . . , Xn in a generic point b. By Proposition 5, we may obtain
π(V ∗(f)) using the system fb. In order to do this, we will compute first a geometric
resolution of V ∗(fb) by means of the subroutine we introduce below.
Let k be a field of characteristic 0. Algorithm ParametricToricGeomRes
computes a geometric resolution of the variety V ∗(g) defined from a generic sparse
system g := (g1, . . . , gr ) in k[X1, . . . , Xt+r ] with given supports S := (S1, . . . , Sr ),
provided that {X1, . . . , Xt } is a set of independent variables for all its components.
This subroutine is obtained by following the parametric geometric resolution
algorithm from [40, Theorem 2] taking X1, . . . , Xt as the parameters.
Algorithm ParametricToricGeomRes
INPUT: A generic sparse system g := (g1, . . . , gr ) in k[X1, . . . , Xt+r ] with given
supports S := (S1, . . . , Sr ) such that {X1, . . . , Xt } is algebraically independent
modulo each associated prime of (g1, . . . , gr ) : (X1 · · · Xt+r )∞.
1. Choose ξ = (ξ1, . . . , ξt ) at random with ξi ∈ Z − {0} for every 1 ≤ i ≤ t .
2. Compute a geometric resolution of the common solutions of g(ξ, Xt+1, . . . , Xt+r )
in (k∗)r .
3. Obtain an slp encoding the polynomials in g.
4. Apply a symbolic Newton–Hensel lifting (in the parameters X1, . . . , Xt ) to the
geometric resolution obtained in step 2 with precision 2MV(S, Δ(t)).
5. Applying Padé approximation to the output of the previous step, recover
numerators and denominators in k[X1, . . . , Xt ] for the coefficients of the polynomials in
the geometric resolution of V ∗(g).
OUTPUT: A geometric resolution of V ∗(g) with free variables X1, . . . , Xt .
Before estimating the complexity of the previous algorithm, we present a simple
example to illustrate how the algorithm works.
Example Let g be the following sparse system supported on S = (S1, S2), where
S1 = {(0, 0, 0), (1, 1, 0), (0, 1, 1)} and S2 = {(0, 0, 0), (2, 1, 1), (0, 2, 0), (1, 1, 1)}:
g :=
g1 = 2 + 3X1X2 − X2X3,
g2 = −1 + 2X12X2X3 + 2X22 + X1X2X3.
Here, {X1} is algebraically independent modulo (g1, g2) : (X1X2X3)∞.
In step 1, the algorithm chooses a value ξ1 ∈ Z − {0} at random and specializes
X1 = ξ1. Setting ξ1 = 1, we obtain the system
g1 :=
g11 = 2 + 3X2 − X2X3,
g21 = −1 + 3X2X3 + 2X22.
Now, the algorithm computes a geometric resolution of the common zeros of g1
in (C∗)2. This is a generic system supported on ({(0, 0), (1, 0), (1, 1)}, {(0, 0), (1, 1),
(2, 0)}). Then, in order to solve it, we can apply the algorithm from [30, Sect. 5].
Choosing X3 as separating linear form, we obtain
• qX3 (Y ) = Y 2 − 152 Y − 51
5 3
• w2(Y ) = − 4 Y − 4
• w3(Y ) = Y
From this geometric resolution, the algorithm computes a sufficiently good
approximation of a geometric resolution of V ∗(g).
A geometric resolution of V ∗(g) consists of univariate polynomials with
coefficients that are rational functions in the parameter X1 with numerators and
denominators of degrees bounded by deg V ∗(g) (see [40, Theorem 1]). These rational
functions can be regarded as power series in the variable X1 centered at ξ1 = 1,
and therefore, they can be recovered by means of Padé approximation from
sufficiently many terms of their expansions. The precision required to do this equals
2 deg V ∗(g) = 2MV(S, Δ). Since MV(S, Δ) = 6, in step 4, the algorithm applies a
Newton–Hensel lifting (in the parameter X1), as explained in [40, Sect. 4.2], to the
geometric resolution (qX3 , w2, w3) with precision 12, obtaining
where
• q1 = − 152 − 158 (X1 − 1) + 2158 (X1 − 1)2 − 2245 (X1 − 1)3 + 116285 (X1 − 1)4 −
4285 (X1 − 1)5 + 1672258 (X1 − 1)6 − 2642956 (X1 − 1)7 + 138102458 (X1 − 1)8 − 236112152 (X1 − 1)9 +
Finally, Padé approximation is applied, following [48, Corollaries 5.21 and 6.48], to
each of the coefficients previously computed in order to obtain the numerators and
denominators of the coefficients in the geometric resolution of V ∗(g) associated to
the linear form X3:
• QX3 (Y ) = Y 2 +
1 − 21 X1 + 41 )Y − 43 X1
• W2(Y ) = (−X2
• W3(Y ) = Y
Now we explain how the different steps of the algorithm are carried out in general
and estimate the number of operations in k it performs. We will use the notation
N = 1≤j ≤r #Sj and d = max1≤j ≤r deg(gj ).
In Step 2, the algorithm computes the sparse encoding of g(ξ , Xt+1, . . . , Xt+r )
within O(r (t + r )N log(d )) operations. Then, this system is solved using the
procedure from [30, Sect. 5] within complexity O(r 3N log(d )M (M)M (MV ( (S ))) ×
(M (MV ( (S ))) + M (ωmax 1≤h≤r MV ( (Sj )j =h, Δ)))), where is the
projection to the last r coordinates, ωmax is the maximum of the values taken by a generic
lifting function ω for (S ) := ( (S1), . . . , (Sr )), and M := max{ μ }, the
maximum ranging over all primitive normal vectors to the mixed cells in the fine mixed
subdivision of (S ) induced by ω.
In Step 3, an slp of length O((t + r )N log(d )) encoding the polynomials g is
obtained from their sparse representation.
The next step is performed modifying the procedure in [40, Sect. 4.2] in
order to use straightline programs for computations with truncated multivariate
power series. We represent each of these series as the vector of its homogeneous
components and these components by means of straightline programs. The
required precision is 2 deg(V ∗(g)) = 2MV (S , Δ(t)) (see [40, Theorem 1] and [25,
Lemma 1]). Therefore, this step is done within complexity O((r (t + r )N log(d ) +
r 4)M (MV ( (S )))MV (S , Δ(t))2) and produces an slp of the same order encoding
the homogeneous components of the coefficients of the output.
Finally, the Padé approximation to obtain the coefficients of the geometric
resolution is done by reducing it to a univariate problem and solving it by
means of subresultant computations following [48, Corollaries 5.21 and 6.48].
This step adds O(rMV (S, Δ(t))Ω+2) operations to the previous complexity and
O(rMV (S, Δ(t))Ω+1) to the slp length.
Therefore, we have the following complexity result:
Lemma 8 Let g := (g1, . . . , gr ) in k[X1, . . . , Xt+r ] be a generic sparse system
with supports S := (S1, . . . , Sr ) such that {X1, . . . , Xt } is algebraically
independent modulo each associated prime of (g1, . . . , gr ) : (X1 · · · Xt+r )∞. Algorithm
ParametricToricGeomRes computes a geometric resolution of V ∗(g). With the
previous notation, the total number of operations in k performed by the algorithm is
of order
O
r3 + rt N log(d)M(M)M MV
(S)
MV S, Δ(t) 2
+ M
ωmax
MV
(Sj )j=h , Δ
+ rMV S, Δ(t) Ω+2 .
The algorithm produces an slp of length
O rMV S, Δ(t) 2 (t + r)N log(d) + r3 M MV
(S)
+ MV S, Δ(t) Ω−1
for the coefficients of the output.
The last step of our main algorithm consists in describing the projection to a
coordinate subspace of an equidimensional variety of dimension t given by a geometric
resolution in the case that the projection has the same dimension t . To do this, we
apply the subroutine described below.
Let V ⊂ kt+r be an equidimensional variety of dimension t definable over k and
such that for each irreducible component W of V , I (W ) ∩ k[X1, . . . , Xt ] = {0} holds.
Consider the projection π : kt+r → k where > t , π(x1, . . . , xt+r ) = (x1, . . . , x ).
Note that {X1, . . . , Xt } are free variables with respect to each irreducible
component of π(V ). Let K := k(X1, . . . , Xt ). Suppose λ ∈ k[Xt+1, . . . , Xt+r ] is a
primitive element for K ⊗ k[V ] and let (qλ, wt+1, . . . , wt+r ) ∈ K[Y ]r+1 be the
associated geometric resolution. Let D be the dimension of K ⊗ k[V ] as Kvector
space.
Let μ = μt+1Xt+1 + · · · + μ X be a primitive element for K ⊗ k[π(V )]. As
I (π(V )) = I (V ) ∩ k[X1, . . . , X ], to find the minimal polynomial of μ with
respect to π(V ), it suffices to find a polynomial qμ ∈ K[Y ] of minimal degree such
that qμ(μ) ∈ K ⊗ I (V ). Then, δ := degY (qμ) is the dimension of K ⊗ k[π(V )] as
a Kvector space and so, for each t + 1 ≤ j ≤ , in order to obtain a polynomial
vj such that Xj = vj (μ) in K ⊗ k[π(V )] it suffices to find a linear combination
Xj = iδ=−01 vji μi of {1, μ, . . . , μδ−1} in K ⊗ k[V ]. To do this, we use the basis
Bλ := {1, λ, . . . , λD−1} of K ⊗ k[V ].
In order to compute the geometric resolution of π(V ) associated with μ, we first
look for the minimal power μδ which is a Klinear combination of {1, μ, . . . , μδ−1}
in K ⊗ k[V ]. Since Xj = wj (λ) for every t + 1 ≤ j ≤ t + r , we have μ =
j=t+1 μj wj (λ) = pμ(λ), where pμ(Y ) := j=t+1 μj wj (Y ) and, for every i ∈ N,
μ = pμ(λ)i = (pμ(Y )i mod qλ(Y ))Y =λ. Therefore, δ equals the rank of the D × D
i
matrix whose columns are the coefficients of the polynomials (pμ(Y )i mod qλ(Y ))
for i = 0, . . . , D − 1.
Then, we obtain the minimal polynomial qμ(Y ) := Y δ + iδ=−01 qμ,i Y i of μ and
the polynomials vj (Y ) = iδ=−01 vji Y i , t + 1 ≤ j ≤ , which form the geometric
resolution of π(V ), by solving the linear systems obtained by equating the coefficients
of the different powers of λ in the identities
δ−1
i=0
pμδ (λ) =
(−qμ,i )pμi (λ) and
wj (λ) =
vji pμi (λ), t + 1 ≤ j ≤ .
Summarizing, with the previous notation and hypothesis, we have the following.
δ−1
i=0
Algorithm GeomResProj
primitive element for K ⊗ k[π(V )].
INPUT: A geometric resolution (qλ, wt+1, . . . , wt+r ) of V with free variables
X1, . . . , Xt , and a linear form μ = j−=t1 μt+j Xt+j ∈ k[Xt+1, . . . , X ] which is a
1. Set pμ0 (Y ) := 1 and pμ(Y ) := hD=−01( j=t+1 μj wj,h)Y h, where (wj,0, . . . ,
wj,D−1) =: wj is the vector of coefficients of wj (Y ) for j = t + 1, . . . , t + r .
2. For i = 2, . . . , D, compute pμi (Y ) := (pμ(Y ) · pμi−1 (Y ) mod qλ(Y )).
3. Compute δ := rank(pμ0 , pμ, . . . , pμD−1 ), where pμi ∈ KD×1 denotes the vector of
coefficients of pμi .
4. Set P := (pμ0 , pμ, . . . , pμδ−1 ) ∈ KD×δ .
5. Solve the linear systems P · q = pμδ and P · vj = wj , t + 1 ≤ j ≤ , to obtain
q := (q0, . . . , qδ−1) and vj := (vj,0, . . . , vj,δ−1).
6. Set qμ(Y ) := Y δ − iδ=−01 qi Y i and vj (Y ) := iδ=−01 vji Y i for every t + 1 ≤ j ≤ .
OUTPUT: The geometric resolution (qμ, vt+1, . . . , v ) of the projection π(V ) ⊂ k
with free variables X1, . . . , Xt associated to the linear form μ.
The correctness of the procedure follows from our previous arguments. Now we
estimate its complexity. Assume that the input polynomials are encoded in dense
form as degree D univariate polynomials in k(X1, . . . , Xt )[Y ] and their coefficients
are encoded by an slp over k of length L.
In Step 1, the algorithm computes an slp encoding the coefficients of pμ of length
bounded by L + 2D( − t ).
In order to fulfill Step 2, we first compute recursively the powers Y h for h =
D, . . . , 2D − 2 modulo qλ(Y ) from the coefficients of qλ and then, we obtain an
slp of length O(L + D( − t ) + D3) for the coefficients of pμi for i = 2, . . . , D by
expanding the product pμ(Y ) · pμi−1 (Y ) and substituting the powers of Y by their
previously computed expressions.
Step 3 is done probabilistically by choosing a point x = (x1, . . . , xt ) at random,
evaluating the involved rational functions at this point within O(L + D( − t ) + D3)
operations in k and finally computing the rank δ of (pμ0 (x), pμ(x), . . . , pμD−1 (x))
within O(Dω) operations in k, with ω < 3 (see [5, Chap. 2, Sect. 2]).
To solve the linear systems involved in Step 5, the algorithm computes the
invertible matrix Pt P within O(Dδ2) operations in k, its adjoint matrix and determinant
within O(δΩ ) additional operations, and the products adj(Pt P)pμδ and adj(Pt P)wj
for t + 1 ≤ j ≤ with O(δ2( − t )).
Adding the previous estimates, we conclude that the algorithm produces an slp of
length O(L + D3 + Dδ( − t ) + δΩ ) over k within a complexity of the same order.
Taking into account that δ ≤ D, we have the following.
Lemma 9 Let V ⊂ kt+r be an equidimensional variety of dimension t definable over
k and such that for each irreducible component W of V , I (W ) ∩ k[X1, . . . , Xt ] = {0}
holds. With the previous notation, given a geometric resolution of V with free
variables X1, . . . , Xt , and a linear form μ which is a primitive element for K ⊗ k[π(V )],
Algorithm GeomResProj computes the geometric resolution of the projection
π(V ) ⊂ k with free variables X1, . . . , Xt associated to the linear form μ within
O(L + DΩ + D2( − t )) operations in k. The output of the algorithm is encoded by
an slp of length of the same order.
4.2 An Algorithm to Find the Projection
Here we present a probabilistic algorithm that, from a fixed family A = (A1, . . . , Ar )
of finite sets (Z≥0)n and a fixed family of variables X1, . . . , X , obtains a
geometric resolution of π(V ∗(f)) ⊂ K , where f = (f1, . . . , fr ) is defined in Eq. (1) and
π : Kn → K is the projection π(x1, . . . , xn) = (x1, . . . , x ).
Algorithm K Projection
INPUT: A family A = (A1, . . . , Ar ) of finite subsets of (Z≥0)n such that
dim( j∈J Aj ) ≥ #J for every J ⊂ {1, . . . , r} and a set of variables {X1, . . . , X } ⊂
{X1, . . . , Xn}.
1. Apply Algorithm TransBasis to the family A. Without loss of generality, we
suppose the transcendence basis obtained is {X1, . . . , Xt , Xt+r+1, . . . , Xn} with
t ≤ and t + r + 1 > .
2. Choose randomly in Z the entries of a vector b = (bt+r+1, . . . , bn) and of a vector
(λt+1, . . . , λt+r ).
3. Obtain the sparse representation of the system of polynomials
fb = (f1(X1, . . . , Xt+r , b), . . . , fr (X1, . . . , Xt+r , b)) in K[X1, . . . , Xt+r ].
4. Apply Algorithm ParametricToricGeomRes to the system fb and the
variables X1, . . . , Xt to obtain the geometric resolution (qλ, wt+1, . . . , wt+r ) of the
variety V ∗(fb) with free variables X1, . . . , Xt associated to the linear form λ =
λt+1Xt+1 + · · · + λt+r Xt+r .
5. Choose randomly in Z the entries of a vector (μt+1, . . . , μ ).
6. Apply Algorithm GeomResProj to (qλ, wt+1, . . . , wt+r ) and μ = μt+1Xt+1 +
· · · + μ X .
OUTPUT: A geometric resolution (qμ, vt+1, . . . , v ) of π(V ∗(f)) ⊂ K , where
π : Kn → K is the projection to the first coordinates.
Theorem 10 Given a family A = (A1, . . . , Ar ) of finite subsets of (Z≥0)n such that
dim( j∈J Aj ) ≥ #J for every J ⊂ {1, . . . , r} and the projection π : Kn → K to
the first coordinates, Algorithm K Projection is a probabilistic procedure that
computes a geometric resolution of π(V ∗(f)) for the sparse system f supported on A
with indeterminate coefficients within
O n2 + r3 N log(d)M(D)Ξ D2 + M(E ) + rDΩ+2
operations in K , where N = 1≤j≤r #Aj , d := max1≤j≤r {degX(fj )}, D =
MV(A, Δ(n−r)), E = 1≤h≤r MV(((Aj )j=h), Δ(n−r+1)), and Ξ is a constant
measuring the size of certain combinatorial objects involved at intermediate
computations and associated to the family of supports A.
Proof Since in our complexity estimates we only take into account the number of
operations in K (and not mixed volume or mixed subdivision computations), to obtain
the overall complexity of the algorithm it suffices to add the complexities of steps 3,
4 and 6.
Step 3 can be done within O(n2N log(d)) operations in K .
The complexity of Step 4 is the already stated for Algorithm
ParametricToricGeomRes in Lemma 8. Note that for generic b, the system fb is a generic
polynomial system supported on S := (S1, . . . , Sr ), where Sj ⊂ (Z≥0)t+r is the
projection of Aj to the first t + r coordinates for every 1 ≤ j ≤ r . Moreover, by
Bernstein’s Theorem, MV(S, Δ(t)) ≤ MV(A, Δ(n−r)), MV( (S)) ≤ MV(A, Δ(n−r))
and, for every 1 ≤ h ≤ r , MV( ((Sj )j=h), Δ) ≤ MV(((Aj )j=h), Δ(n−r+1)). We
take Ξ such that M(M)M(ωmax) ≤ Ξ .
Finally, the complexity of Step 6 follows from Lemma 9. Note that, here, D ≤
MV(A, Δ(n−r)) and L is the length of the slp computed in Step 4 according to
Lemma 8.
4.3 Example
Consider a sparse system of two polynomials in five variables supported on A =
(A1, A2), where A1 = {(0, 0, 0, 0, 0), (1, 1, 1, 0, 0), (2, 0, 0, 4, 2), (0, 0, 0, 8, 4)} and
A2 = {(1, 0, 1, 1, 2), (0, 1, 2, 5, 4), (1, 3, 0, 5, 4)}, with indeterminate coefficients:
f :=
f1 = C11 + C12X1X2X3 + C13X12X44X52 + C14X48X4,
5
f2 = C21X1X3X4X52 + C22X2X32X45X54 + C23X1X23X45X54
and the projection π : A5 → A3, π(x1, x2, x3, x4, x5) = (x1, x2, x3). We are going to
show a geometric resolution of π(V ∗(f)) following Algorithm K Projection.
First, we apply Algorithm TransBasis and we find that {X1, X2, X4} is a
transcendence basis of K(V ∗(f)), and so, {X1, X2} is a transcendence basis of
K(π(V ∗(f))).
Now, the algorithm chooses a value b at random and specializes X4 = b. Set b = 1.
The specialized system is
f1 :=
f11 = C11 + C12X1X2X3 + C13X12X52 + C14X54,
f21 = C21X1X3X52 + C22X2X32X54 + C23X1X23X54.
The next step is to apply Algorithm ParametricToricGeomRes with free
variables X1, X2. We choose λ = X5 as the primitive element to obtain the geometric
resolution:
• qX5 (Y ) = Y 10 + 2CC1134X12 Y 8 + C123X14+C1224C11C14 Y 6 +
−C12C21C13X14+C121C22+C122C23X13X24 Y 2 C12C21C11X12
C22C124 − C22C124
• w3(Y ) = − C12CX141X2 Y 4 − CC1132XX12 Y 2 − C12CX111X2
• w5(Y ) = Y
(−C12C21C14+2C11C22C13)X12 Y 4
C22C124
+
Finally, Algorithm GeomResProj is applied to qX5 , w3, w5 and a primitive
element μ for K(π(V ∗(f))). In this case, we take μ = X3 and obtain
+
• v3(Y ) = Y
C11
• qX3 (Y ) = Y 5 + C12X1X2 Y 4 +
4.4 Computation for Generic Rational Coefficients
2C12C23X1X24−C13C21X12 Y 3 + 2C23C22C11X24+C221C14X1 Y 2
3
C12C222X2
We are now going to show that, for a system with generic rational coefficients, the
same steps as the ones in Algorithm K Projection can be performed using these
rational coefficients to obtain a geometric resolution of the Zariski closure of the
projection of the associated variety.
Let {X1, . . . , Xt , Xt+r+1, . . . , Xn} be a transcendence basis of K(V ∗(f)) such that
{X1, . . . , Xt } is a maximal algebraically independent subset of {X1, . . . , X }.
As shown in [11, Appendix A], there exists a nonempty Zariski open set U1 ⊂ AN
such that if c = (c1, . . . , cr ) ∈ U1 ∩ QN , then:
• the ideal Ic := (f1(c, X), . . . , fr (c, X)) : (X1 · · · Xn)∞ is radical equidimensional
of dimension n − r ,
Assume c ∈ U1 ∩ QN and consider the variety V ∗(f(c)) ⊂ An. Let W be an
irreducible component of V ∗(f(c)). We have dim(W ) = n − r and {X1, . . . , Xt ,
Xt+r+1, . . . , Xn} is a transcendence basis of Q(W ). Then, the projection of W over
the last n − t − r coordinates is a dominant map. Therefore, there is a Zariski open
subset OW ⊂ An−t−r such that, for every b ∈ OW ∩ Qn−t−r :
• Wb := W ∩ {xt+r+1 = bt+r+1, . . . , xn = bn} is an equidimensional variety of
dimension t ,
• {X1, . . . , Xt } is algebraically independent modulo I (Wb).
Then, for every b ∈ OW ∩ Qn−t−r , the identity π(W ) = π(Wb) holds.
As the dimension of the set ∂(f(c)) := V ∗(f(c)) − {x ∈ (C∗)n : f(c, x) = 0} is less
than n − r , for every {i1, . . . , it } ⊂ {1, . . . , t + r}, there exists a nonzero
polynomial pi1,...,it (Xi1 , . . . , Xit , Xt+r+1, . . . , Xn) vanishing identically on this set. Then,
there is a nonempty Zariski open set O1 ⊂ An−t−r such that for every b ∈ O1, the
dimension of ∂(f(c)) ∩ {xt+r+1 = bt+r+1, . . . , xn = bn} is less than t .
Then, for every b ∈ O1 ∩ W OW ∩ (Q∗)n−t−r , we have
V ∗ f(c) ∩ {xt+r+1 = bt+r+1, . . . , xn = bn} = xˆ ∈
C∗ t+r : f(c, xˆ, b) = 0 × {b}
and π(V ∗(f(c))) =
xn = bn}); therefore,
W π(W ) =
W π(Wb) = π(V ∗(f(c)) ∩ {xt+r+1 = bt+r+1, . . . ,
π V ∗ f(c)
= π
xˆ ∈
C∗ t+r : f(c, xˆ, b) = 0 × {b} .
For j = 1, . . . , r , let Sj ⊂ (Z≥0)t+r be the projection of Aj to the first t + r
coordinates. Then fj = aˆ∈Sj ( (aˆ,a˜)∈Aj Cj,(aˆ,a˜)X˜ a˜ )Xˆ aˆ . Algorithm
ParametricToricGeomRes works for generic sparse polynomial systems g supported on
S = (S1, . . . , Sr ), that is, there is a polynomial pS in the coefficients of the system
such that it computes a geometric resolution of V ∗(g(cˆ)) for every vector of
coefficients cˆ with pS (c) = 0. Let U2 ⊂ AN be a nonempty Zariski open set such that, for
ˆ
every c := (c1, . . . , cr ), the polynomial pS (( (aˆ,a˜)∈Aj cj,(aˆ,a˜)X˜ a˜ )1≤j≤r,aˆ∈Sj ) does
not vanish identically.
For c ∈ U2 ∩ QN , there exists a nonempty Zariski open set O2 ⊂ An−t−r such
that for every b ∈ O2 ∩ Qn−t−r , the algorithm ParametricToricGeomRes can
be applied to the system f(c, xˆ, b).
We conclude that, for coefficient vectors c ∈ U1 ∩ U2 ∩ (Q∗)N , a
probabilistic algorithm QProjection which follows the same steps as Algorithm K
Projection can be applied in order to compute a geometric resolution of
π(V ∗(f(c))). Taking into account the complexity estimates in Theorem 10, this
proves Theorem 1.
Finally, we show an example where, following the steps of the algorithm
QProjection, we obtain a geometric resolution of π(V ∗(f)) for a sparse system
f with rational coefficients.
Example Let f be the following sparse system with the same support family A =
(A1, A2) as in the example of Sect. 4.3:
f =
In Step 6, if Algorithm GeomResProj is applied to the geometric resolution
obtained in Step 4 and the linear form μ, the geometric resolution (qX3 , v3) is obtained,
where
Acknowledgements The authors thank the referees for their helpful comments and suggestions. This
work was partially supported by the Argentinian research grants CONICET PIP 0099/11, UBACYT
20020090100069 (2010–2013) and UBACYT 20020120100133 (2013–2016).
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