Matrices in Elimination Theory

Ioannis Z Emiris Bernard Mourrain

INRIA SAGA BP

SophiaAntip olis France

FirstLastsophiainriafr

httpwwwinriafrsaga Fir stL ast

Septemb er

Contents

Intro duction

Elimination theory

Notation

Classical elimination theory

Resultant over P

Resultant over a toric variety

Matrix formulations

Macaulay matrices

Newton matrices

Incremental Newton matrices

Bzout matrices

Dixon matrices

Comparison b etween dierent matrices

Applications of resultant matrices

Monomial bases and multiplication maps

System solving by the uresultant

System solving by eigenvectors

Genericity issues

Conclusion

Abstract

The last decade has witnessed the rebirth of resultant metho ds as a p owerful computational to ol for

variable elimination and p olynomial system solving In particular the advent of sparse elimination theory

and toric varieties has provided ways to exploit the structure of p olynomials encountered in a numb er of

scientic and engineering applications On the other hand the Bezoutian reveals itself as an imp ortant

to ol in many areas connected to elimination theory and has its own merits leading to new developments

in eective algebraic geometry This survey unies the existing work on resultants with emphasis on

constructing matrices that generalize the classic matrices named after Sylvester Bzout and Macaulay

The prop erties of the dierent matrix formulations are presented including some complexity issues

with an emphasis on variable elimination theory We compare toric resultant matrices to Macaulays

matrix and further conjecture the generalization of Macaulays exact rational expression for the resultant

p olynomial to the toric case A sparse version of an eective Nullstellensatz is directly obtained A new

theorem proves that the maximal minor of a Bzout matrix is a nontrivial multiple of the resultant

We discuss applications to constructing monomial bases of quotient rings and multiplication maps as

well as to system solving by linear algebra op erations Lastly degeneracy issues a ma jor preo ccupation

in practice are examined Throughout the presentation examples are used for illustration and op en

questions are stated in order to p oint the way to further research

Intro duction

Resultants provide an essential to ol in constructive algebra and in equation solving The resultant of an

overconstrained p olynomial system characterizes the existence of common ro ots as a condition on the input

co ecients If we consider the input co ecients as indep endent indeterminates then the solutions lie in a

space of dimension n m where n is the numb er of variables and m is the numb er of co ecients The

resultant pro jects the solutions to an mdimensional space and is therefore also known as a projection

operator Since it eliminates the input variables the resultant is also known as the eliminant of the given

system

A numb er of metho ds exists for constructing resultant matrices which are matrices whose determinant

is the resultant or more generally a nontrivial multiple of it These matrices represent the most ecient

way for computing the resultant p olynomial and for solving systems of p olynomial equations by means of

the resultant metho d An example of a matrix that gives precisely the resultant is the determinant of the

co ecient matrix of n linear p olynomials or the Sylvester matrix of a p olynomial pair Resultant matrices

have b een extensively studied around the turn of the century Their determinants have b een known as inertia

forms We give a short historical overview of the impact of resultants in eective algebraic geometry We

refer the reader to Mui for a more complete historical description

The rst ma jor contribution to resultant theory was probably the work of E Bzout B and Euler

By combining two univariate p olynomials P x of degree m Qx of degree n m Bezout observed that he

can get m linearly indep endent p olynomials of degree m This yields a m by m matrix called Bezoutian

matrix by Sylvester whose determinant is the resultant of P and Q originally called la rsultant de P et

Q in French see Las It was worked out latter by Cayley who connected it with the p olynomial

Qx P y Qy P x

x y

JJ Sylvester preferred to eliminate directly the monomials x x in the multiples

i j

x P x x Qx

in j m

of the initial p olynomials and to ok the determinant of the corresp onding m n m n matrix Although

contemp orary and related works Jacobi Richelot Cauchy this metho d remain well

known as Sylvesters resultant

A generalization of the Bezoutian in several variables was used in the work of A Dixon Dix For

p olynomials P P P in two variables of the same degree he to ok some co ecients of their multivariate

Bezoutian and added some multiples of the initial p olynomials in order to get a square matrix In section

we shall come again to this construction

The work of FS Macaulay see Mac Mac vdW generalizes the Sylvester construction to the

multivariate case which in turn was extended recently to the case of toric variety Indeed the last two

decades have witnessed the ourishing of the theory of sparse elimination Ber GKZ a more complete

account is given b elow This theory generalizes several results of classical elimination theory on multivariate

p olynomial systems by considering the structure of the given p olynomials namely the co ecients which are a

priori zero and the supp ort and Newton p olytop e dened by the nonzero co ecients This leads to stronger

algebraic and combinatorial results in general whose complexity dep ends on eective rather than total

degree The toric or sparse resultant generalizes the classical resultant of n homogeneous p olynomials

in n variables in the sense that they coincide when all p olynomial co ecients are nonzero The toric

resultant coincides with the Sylvester resultant if the system is comprised of two univariate p olynomials

Unlike its classical counterpart however the toric resultant dep ends on the nonzero monomials only and

therefore it has lower degree for sparse inputs

The renewed interest in elimination theory and the asso ciated matrix metho ds for system solving is man

ifold Bezoutians app ear to b e a fundamental to ol in several domains such as residue theory SS Kun

and complex analysis BGVY BY AK Dierent kinds of resultant matrices are used in com

plexity theory in eective algebraic geometry SS FGS LM and of course elimination theory

Jou The use of Bezoutians in imp ortant applications from an algorithmic p oint of view is illus

trated in Car BCRS CM For small and mediumsize p olynomial systems corresp onding to zero

dimensional varieties the resultant matrix provides one of the most ecient solution metho ds to day This

has b een established through a numb er of concrete applications in the forward and inverse kinematics of

rob ots and mechanisms as well as the computation of their motion plans Can RR the geometric

structure of molecules BMB EMb geometric and solid mo deling graphics and computeraided de

sign BGW Hof MD as well as quantier elimination CG Ren Can BPR and the

solution of systems of inequalities GV

This survey is organized as follows The next section sketches the basic notions of elimination theory

starting with notation then the classical theory and nally discusses sparse elimination theory Section is

the main part where the dierent matrix formulations are detailed and analyzed Sylvester and Macaulay

matrices in section two algorithms for the toric resultant or Newton matrix in sections and

Bzout and Dixon matrices in the following two sections and nally a comparison b etween dierent matrix

formulations Then we consider how these matrices can b e used for constructing monomial bases multiplica

tion maps and ultimately for solving systems of p olynomial equations by dierent metho ds in sections

to Section presents metho ds for handling input degeneracy We conclude with a summary

Elimination theory

Notation

K is the co ecient eld unless otherwise stated it is arbitrary P denotes the pro jective space

K is the of dimension n obtained as a quotient of K nonzero multiple vectors are identied

algebraic closure of K K K fg is the eld K except

n is the numb er of variables of the p olynomial rings of interest x x denote the n indep endent

variables and x is a shorthand for all of them The notation x represents x x More sets

of variables are sometimes needed and denoted y and u

The set of p olynomials in the variables x x with co ecients in K will b e denoted by R

K x x f f are the input p olynomials lying in R K x These can b e Laurent

n n

p olynomials lying in R K x x Their co ecients are denoted c c

I J are ideals of R and A R I denotes the quotient algebra of p olynomials p mo dulo the ideal

I Z f f stands for the zeroset of f f that is the algebraic variety of p oints

K s s

K such that f f If not sp ecied Z f f means the zeroset over the

s s

K of K The set of ro ots of f f which are in an algebraic variety X is denoted algebraic closure

by Z f f

X s

M is the resultant matrix Occasionally it stands for a candidate matrix or the rectangular matrix

from which a square matrix is obtained I is the identity matrix whose dimension is clear from context

or sp ecied by I if it equals k

Q for i n denote the Newton p olytop es of the given p olynomials dened in section

P P

Q is the Q Q is the Minkowski sum of the given n Newton p olytop es and Q

j i i

Minkowski sum of n of them intro duced in section

n n

Q Z E is the integer p oint set Q Z For i n E is the p oint set

j i

Q is the p erturbation vector used in the sub divisionbased algorithms of section is a

real p ositive innitesimal used in dierent contexts so v is a vector in Q used in the

incremental algorithm in section

V is the standard Euclidean volume function

MV denotes the mixed volume op eration on p olytop es Given a set of n p olytop es MV for

i n denotes the mixed volume of n Newton p olytop es excluding the ith one When

these op erators are applied to p olynomials or p oint sets we understand the mixed volume of the

corresp onding Newton p olytop es or resp ectively the convex hulls of the p oints These op erators are

dened in section

Classical elimination theory

Elimination theory deals with the problem of nding conditions on parameters of a p olynomial system so

that these equations have a common solution in an algebraic set that we denote hereafter by X A typical

situation is the case of n p olynomials

f x c x

j j

n+1

f x c x

n nj nj

where

c c are parameters

x is a p oint of the pro jective variety X P of dimension n

the functions x j k are homogeneous p olynomials indep endent of the parameters c

ij i

and of the same degree in the co ordinates of x P

and by f x the Let us denote by L x the vector of p olynomial functions L x x

c i i ij j k

global system of equations

The elimination problem consists in this case in nding necessary and sucient conditions on the

parameters c c such that the equations f f have a common ro ot in X Note

ij ij n

that if the numb er of equations is not greater that the dimension of X then there is no condition on the

parameters This is the reason why we cho ose X of dimension n

is the vector of all monomials of degree The classical situation is the case where L x x

i ij j k

d and where X P is the pro jective space of dimension n The functions f x are generic homogeneous

i i

p olynomials of degree d and the necessary and sucient condition on the parameters c c such that

i ij ij

f f where the homogeneous p olynomials f f have a common ro ot in X P is Res

n n P

Res is the classical projective resultant

Considering a geometric p oint of view we are lo oking for the set of parameters c c such that there

c x for i n In other words the parameter vector c is the exists x X with

ij ij

pro jection of the p oint c x of the variety

k k

n+1 1

c x i n g X st P W fc x P

ij ij X

This variety W is called the incidence variety and we have two pro jections

k k

n+1 1

P W P

W X

The image of W by is precisely the set of parameters c for which the system has a ro ot The image by

of a p oint of W is a solution in X of the asso ciated system Any p olynomial in c c which vanishes

X ij ij

on the pro jection W is called an inertia form see vdW The inertia forms are homogeneous

of parameters p olynomials in each subset c

ij j k

Ane algebraic sets dened by p olynomial equations may pro ject onto sets dened by equalities and

inequalities eg the hyp erb ola dened by x y pro jects onto the xaxis except But pro jective

This notation refers to line bundles as we will see in the following

varieties pro ject onto pro jective varieties dened by homogeneous p olynomial equations see Har

Sha Their pro jection is closed for the Zarisky top ology That is the reason why we assume here that

X is a subvariety of a projective space Moreover we will assume that X is irreducible for we can reduce

Notice that the variety W dened by multihomogeneous W W our study to this case W

X X X X X

2 1 1 2

k k

n 0

P equations is also a pro jective variety Therefore its pro jection by is a closed subvariety of P

In many case the variety X do es not app ear explicitly in the resultant formulation problem but is given

N n

implicitly as the closure in P of a parameterized ane variety X A This is the case for instance for

sparse elimination theory as we will see in the next section Let us denote by T X T is a dense

X is necessarily irreducible see subset of A the rational map which parameterizes X The variety X

Har

Denition Let Z W If Z is an hypersurface then its equation unique up to a scalar wil l be

cal led the resultant of f f on X It wil l be denoted by Res f f

n X n

In other words when co dimZ Res is dened up to a scaling and f x f x have a

X n

solution in X if and only if Res f f This generalizes the denition of the classical resultant

X n

over P to any irreducible pro jective variety

In order to b e in the case co dimZ we imp ose the following conditions

Condition

For any point x X and any i n the vector L x is not zero

For generic value of the parameters c the system f x has no solution in X

The rst condition is needed to derive easily the prop erties on W The second condition is there to

avoid degenerate cases such as L L for i

Prop osition Under conditions the projection Z is of codimension and the resultant Res of

f f on X is uniquely dened up to a scaling It is an irreducible polynomial of Zc

n ij

Pro of As for any x X and any i n the vector L x is nonzero the set x is a linear

k k

n+1 1

fxg of dimension k n Consequently by the b er theorem see P space of P

Shap Harp we deduce that W is irreducible and of dimension k

X i

Thus its pro jection Z W is an irreducible variety of dimension k or equivalently of

X i

co dimension

k k

n 0

such that f x has no solution on X Let U b e P Let U b e the set of parameters c P

the set of parameters c such that the system f f has a nite numb er of solutions in X

Note that U U for if the solution set of f x f is of dimension then the solution set

k k

0 n

of f x is of dimension By condition U and therefore U are dense subsets of P P

Then W U X is a dense subset of W and pro jects by onto Z U As Z f f is

X X n

nite for any c Z U c fc Z f f Z f g is nite Therefore

X n X

k k

n 0

dened by a unique P W and Z are of the same dimension and Z is an hyp ersurface of P

equation Res f f up to a scaling As Z is irreducible this p olynomial is irreducible and as the

X n

equations dening W are in Zc x Res Zc 2

X ij X ij

In the language of mo dern algebraic geometry the vectors L x dene line bundles on X If the line

bundles are ample then the conditions are satised see GKZ

As a corollary of this condition we obtain the following prop erty that will b e used afterwards Under

condition there exists an op en subset of the set of all co ecients c c such that the system obtained

by removing one of the equations say f has a numb er of isolated ro ots in X which is indep endent of c The

generic numb er of ro ots obtained by removing f is denoted by D f This numb er of ro ots counted with

i i c

multiplicity is the degree of the resultant Res f f in the co ecients of f for i n

X n i

Resultant over P

To illustrate the previous developments we consider here the classical case X P The p olynomials f are

generic homogeneous p olynomials of degree d

c x i n x f

ia i

a a d

0 n i

d d

i i

Thus the vector L is up to a p ermutation L x x x x that is the vector of all monomials

i i

of degree d in the variables x x We easily check that for generic values of c c the system has

i n ia

no solution in P and that L x if and only if x Therefore according to prop osition the

f f is welldened up to a scalar and vanishes if and only if the p olynomials resultant Res

n P

f f have a common ro ot in P

We conclude the section with some fundamental prop erties of the classical resultant discussed in the

aforementioned references

The resultant is an irreducible p olynomial in the variables c c with integer co ecients prop o

sition

The classical resultant is invariant under linear transformations of the variables invariance of the

elimination problem by a change of co ordinates

If f is written as a p olynomial pro duct f f then the resultant is also factored into the cor

n n

n n n

f f f Notice f f f Res f f f Res resp onding pro duct Res

n n P n n P P

n n

that this do es not contradict irreducibility since the co ecients of f are no longer free parameters

and f They are sums of pro ducts of the co ecients of f

n n

The latter prop erty generalizes to the case of a system comprised of p olynomials in an ideal yielding

a divisibility prop erty

n n

f f b b j Res f f b b Res

n n P n n P

The seeming paradox is explained again by the nongenericity of the co ecients

Resultant over a toric variety

We move now to the case of a toric variety X As we will see this variety X is not given explicitly as in

the previous section but dened implicitly by the monomials that actually app ear in the p olynomials f

The p ossibility to tune the resultant construction to the actual set of monomials in the equations is one of

the strength of the theory but it makes it also more dicult to understand We will sketch here the main

results and refer to the work of Gelfand Kapranov and Zelevinsky GKZ for a complete treatment

Polytop es

Elimination theory on a toric variety also called sparse elimination theory considers Laurent p olyno

mials in n variables where the exp onents are allowed to b e arbitrary integers The p olynomial ring is

K x x x x K x x for some base eld K

Denition Let f be a polynomial in K x x The nite set A Z of al l monomial exponents

corresponding to nonzero coecients is the supp ort of f The Newton p olytop e of f is the convex hul l of

A denoted Q convA R

e n e

If we use x to denote the monomial x where e e e Z is an exp onent vector then x

f c c x

j j

a A

The word sparse is ambiguous in this context for it refers to p olynomials which could b e dense with resp ect to their

p olytop es y

Figure The supp ort and Newton p olytop e of p olynomial c y c x y c x y c x c xy The dotted

triangle is the Newton p olytop e of the completely dense p olynomial of the same total degree

Fig depicts the supp ort and Newton p olytop e for a bivariate p olynomial and compares it with the

Newton p olytop e of the completely dense p olynomial with the same total degree ie a p olynomial in which

every co ecient is nonzero Newton p olytop es provide a bridge from algebra to geometry since they p ermit

certain algebraic problems to b e cast in geometric terms For background information on p olytop e theory

and any unproved prop ertied of mixed volumes the reader may refer to Gr Sch

Mixed volume

n n

The Minkowski sum A B of convex p olytop es A and B in R is the set A B fa b j a A b B g R

also denoted by A B A B is a convex p olytop e

Denition Given convex polytopes Q Q R there is a unique up to multiplication by a scalar

realvalued function MV Q Q cal led the mixed volume of the given polytopes which is multilinear

with respect to Minkowski addition and scalar multiplication ie for R and convex polytope Q R

MV Q Q Q Q MV Q Q Q MV Q Q Q

k n k n n

k k

To dene mixed volume completely we require that

MV Q Q n V Q

where V is the Euclidean ndimensional volume function that assigns the unit volume to the hypercube of

unit edge length

Mixed volume generalizes the standard volume function on a single p olytop e Indeed it is the multilinear

function asso ciated with the Volume function It has b een extensively studied by combinatorial geometers

though its denition sometimes diers by a factor of n

An equivalent denition particularly interesting from a computational p oint of view is sp ecied b elow

The mixed of p olytop es Q Q is in some sense the multilinear part in the volume of the sum Q

Q Q Q The idea is to sub divide the p olytop e Q in an union of p olytop es which are sums of

faces of the p olytop es Q We will keep those p olytop es which contribute multilinearly to the volume that

is those which are sums of edges of the p olytop es Q and compute the sum of their volumes

n n

Note that the op eration of Minkowski addition on n p olytop es is a manytoone function from R onto

R mapping an ntuple of p olytop es Q into their Minkowski sum by sending an ntuple of p oints p Q

i i i

into their vector sum

Q Q Q Q

i n

a a a

i n

To sub divide the p olytop e Q we dene a section of this map ie a unique tuple for every p oint in

Q using a standard lifting metho d Below we describ e an ecient version of this technique intro duced

in BS and which is used in several algorithms in toric elimination Select n generic linear lifting forms

l R R i n Then dene the lifted p olytop es

Q fp l p p Q g R i n

i i i i i i

Their Minkowski sum is an n dimensional p olyhedral complex whereas its lower envelop e is an n

dimensional p olyhedral complex dened as the union of all ndimensional faces or facets whose inner

normal vector has p ositive last comp onent The genericity of the l ensures that the lower envelop e pro jects

Q of the original p olytop es Moreover it ensures that every lower bijectively onto the Minkowski sum

b b b

envelop e facet is a unique sum of faces F from the Q for i n such that dim F n

i i i

The sub division of the lower envelop e into facets induces a sub division of Q into cells by pro jecting

each facet onto an ndimensional cell These are maximal cells whereas cells of dimension k n are dened

as the pro jection of lower envelop e faces of dimension k This cell complex is a mixed subdivision Every

maximal cell is a unique Minkowski sum of faces F Q where each F corresp onds to F that app ears

i i i i

in the unique sum dening the corresp onding lower envelop e facet This Minkowski sum is said to b e

optimal since it minimizes the value of the aggregate lifting function over all p ossible ntuples of faces

whose Minkowski sum equals the given cell Due to the linearity of the lifting functions l dim F dim F

i i i

therefore dim F n We dene the mixed cel ls to b e precisely those where all summand faces are

onedimensional ie all F are edges We are now ready to state an prop erty or equivalent denition of

mixed volume BS

Prop osition The sum of the ndimensional Euclidean volumes of al l mixed cel ls in a mixed subdivision

Q is the mixed volume MV Q Q of the given polytopes Q Q R of

i n n

The shorthands MV f f and MV A A are o ccasionally used for the mixed volume MV Q

n n

BKK b ound

The Newton p olytop es oer a convenient mo del for the sparseness of a p olynomial system in light of

Bernsteins upp er b ound on the numb er of common ro ots Ber This b ound is also known as the BKK

b ound to underline the contributions of Kushnirenko and Khovanskii in its development and pro of Kus

Kho

Theorem Let f f K x x x x with Newton polytopes Q Q The number of

n n n

K multiplicities counted is either innite or does not exceed MV Q Q isolated common zeros in

where K is the algebraic closure of K For almost al l specializations of the coecients the number of common

zeros is exactly MV Q Q

Indeed this result is not so surprising for it is easy to see that mixed volume b ehaves like the generic numb er

of ro ots if we replace sum of p olytop es by pro duct of p olynomials and integer multiplication of p olytop es

by exp onentiation of p olynomials

Interesting extensions to this theorem concern the weakening of the genericity condition CR Ro j

arbitrary ane ro ots and the case of arbitrary elds including elds of p ositive characteristic Dan HS

Ro jb

The mixed volume is typically signicantly lower than Bzouts b ound which b ounds the numb er of

deg f where deg f is the total degree of f One example is the simple and pro jective solutions by

i i i

generalized eigenproblems on n n matrices Both can b e expressed by systems of n p olynomials of total

degree two Hence the Bzout b ound in b oth cases is while the numb er of solutions is n b ecause

to each eigenvalue corresp ond two right eigenvectors of opp osite sign A mixed volume computation yields

precisely n see eg LWW

The two b ounds coincide for completely dense p olynomials b ecause each Newton p olytop e is an n

dimensional unit simplex scaled by deg f By denition the mixed volume of the dense system is

n n

Y Y

deg f deg f MV S S MV deg f S deg f S

i i n

i i

where S is the unit simplex in R with vertex set f g

Several ecient algorithms exist for computing mixed volume VVC EC VGC LWW The

main idea of all algorithms is to use a lifting in order to apply denition In terms of complexity classes

mixed volume is Pcomplete Ped

A mixed sub division provides not only the mixed volume but also a monomial basis for the co ordinate ring

asso ciated to the ideal of the given p olynomials This is explained in section The same computation also

sp ecies the start system of a homotopy continuation for numerically approximating all common ro ots HS

VGC These homotopies are called sparse b ecause the numb er of paths followed dep ends on the monomial

structure of the system in particular on its mixed volume

Clearly mixed volume captures the inherent complexity of algebraic problems in the context of toric

elimination and thus provides lower b ounds on the complexity of algorithms In dealing with mixed volumes

some fundamental results can b e found in BZ Sch In particular the AleksandrovFenchel inequality

leads to the following b ound Emi MV Q Q nV Q V Q On the other hand

n n

we will see that the Minkowski sum Q Q contains all the required information to study and

solve a system of n generic p olynomials In the forthcoming sections it shall b ecome clear that several

toric elimination algorithms rely on some Minkowski sum of Newton p olytop es It turns out that a crucial

question in deriving outputsensitive upp er b ounds is the relation b etween mixed volume and the volume of

these Minkowski sums

For a set of n or n Newton p olytop es Q dene its scaling factor s to b e the minimum real value so

that Q t s Q for all Q where Q is the p olytop e of minimum Euclidean volume and the t R are

i i i i

arbitrary translation vectors Clearly s and s is nite if and only if all p olytop es have an ane span

of the same dimension Let e denote the basis of natural logarithms and supp ose that V Q for all i

Then Emi

n n n

n n

X X X

e s

n n

MV O Q O e s MV Q Q V Q V

i i n i

i i i

where MV stands for the mixed volume MV Q Q Q Q i n

i i i n

The toric resultant

In toric elimination theory the main ob ject of study is the toric resultant also known as the toric resultant

For a system of n Laurent p olynomials in n variables the toric resultant characterizes the existence of

nontrivial common zeros in a toric variety X A toric variety can b e dened as the closure of the image of

n N

the torus K by a monomial parametrisation in a pro jective space P

N a n a

N 0

t t t P t K t

See Ful or Cox for a more intrinsic construction By denition the image of the torus K denoted

by X is dense in this variety

We asso ciate to a p olytop e Q the toric variety parametrised by all the monomials of this p olytop e

We denote this variety by T In the context of toric elimination theory we will consider the toric variety

X T where Q Q Q and Q is the Newton p olytop e of f i n If the supp orts

Q n i i

Q are equal then we can instead consider T

i Q

Let c c b e the vector of all p olynomial co ecients regarded as indeterminates and let Z b e the

set of co ecients c such that the system f x has a ro ot in X im or equivalently such that there

K with f t Let Z b e the Zariski closure of Z which can also b e dened as the pro jection exists t

W of the incidence variety W see section over the toric variety X T KSZ GKZ Ro ja

X X Q

A technical assumption is that without loss of generality the ane lattice generated by A is n

dimensional This lattice is identied with Z p ossibly after a change of variables which can b e implemented

by computing the appropriate Smiths Normal form Stu Then we have the following theorem see also

GKZ and theorem

A is ndimensional Then the Prop osition PS Assume that the ane lattice generated by

toric resultant Res f f of polynomials f K x x with supports A for i n is

X n i i

wel l dened up to a scalar It is an irreducible polynomial over their coecients c itself with integer

coecients Furthermore the degree of Res f f in the coecients of polynomial f equals MV

X n i i

for i n

Generic p olynomials are sp ecied with resp ect to their supp ort instead of the total degree of the classical

theory Hence the resultant will also b e dened for a set of supp orts assuming all nonzero co ecients are

generic The resultant Res f f will b e denoted by ResA A where A is the Newton

X n n i

p olytop e of f The vanishing of ResA A is a necessary and sucient condition for the existence

i n

of ro ots in the toric variety X T KSZ Ro ja See also Cox and Ro jc sect

Some fundamental prop erties of the toric resultant as p olynomial in the co ecients c are as follows

The toric resultant subsumes the classical resultant in the sense that they coincide if the generic

p olynomials are dense vdW Section expands on this topic

Just as in the classical case when all co ecients are generic the resultant is irreducible for X is

necessarily irreducible This is essentially stated in prop osition

While the classical resultant is invariant under linear transformations of the variables the toric resultant

is invariant under invertible transformations of the variables that preserve the p olynomial supp ort

Stu GKZ

In the case of nongeneric co ecients analogous divisibility prop erties hold as in the case of the classic

resultant In particular when a system of p olynomials lies in the ideal generated by another system

then the latter resultant is divisible by the former resultant

Matrix formulations

The computation of resultants typically relies on obtaining matrices whose determinant is either the exact

resultant p olynomial or more generally a nontrivial multiple of it In addition for solving p olynomial

systems these matrices are sucient since they reduce the given nonlinear problem to a question in linear

algebra The fo cus of this survey are metho ds for constructing such matrices and their prop erties

Resultant matrices can b e classied into two large families though the distinction is not always completely

clear The matrices that generalize Sylvesters and Macaulays formulations shall b e the topic of the next

section Algorithms for constructing matrices for the toric or sparse resultant also known as Newton

matrices are discussed in sections and Matrices of Bzout typ e will b e discussed in the following

section whereas a metho d combining the two approaches will b e discussed in section Finally section

compares some of the dierent formulations

There is a command for computing the Sylvester matrix on most general computer algebra systems

including Axiom Maple Mathematica and Reduce There is also a command for computing the

Bezoutian in most of these systems eg Maple but it seems to b e less used probably b ecause of the

complex denition of the to ol Their generalization to the multivariate case is under implementation in the

Maple package multires and available at httpwwwinriafrsagalogicielsmultireshtml

In the rst following sections we will consider what we call Sylvestertype matrices For two univariate

p olynomials their resultant equals the determinant of the wellknown Sylvester matrix a very widespread

to ol for variable elimination refer to Syl or Knu DL Zip The Sylvestertype matrices generalize

this construction to multivariate p olynomials f f As in the univariate case the matrices that we

construct represent monomial multiples of the p olynomials f

A n

Let hx i K x x b e the set of all Laurent p olynomials in n variables with supp ort A Z that is

A a n

the vector space generated by the monomials x fx a Ag Now x supp orts B B Z and

consider the following linear transformation

B B B

n+1 1

i hx i i hx S hx

g f g g g

i i n

n b

where B is a subset of Z containing the supp ort of all x f for b B i n The resultant

i i

matrices that we consider are precisely the matrix of such transformations and to dene them fully we have

to sp ecify supp orts B In the next sections we will describ e several formulations sp ecifying these supp orts

We ll in the matrix entries as follows Every column of S is indexed by an element of some B

i n and every row by an element of B equivalently the columns and rows are indexed resp ectively

by the monomials of g and the monomials of g The co ecient in the row corresp onding to b B and in the

b b

columns corresp onding to b B is the co ecient of x in the p olynomial x f Co ecient of monomials

j j

which do not explicitly app ear in x f have a zero entry In other words the column corresp onding to

b B

b B is the co ecient vector of x f in the basis x

j j

Thus the matrix S can b e divided in blo cks S S S each S dep ending only on the co ecients

n i

of the p olynomial f In the case n we recover the usual Sylvester matrix whose blo ck S represents the

d d

2 1

multiples f x x f x x f x and the blo ck S the multiples f x x f x x f x

The numb er of columns equals the sum of the cardinalities of supp orts B while the numb er of rows

equals the cardinality of B In the sequel we restrict ourselves to matrices S with at least as many columns

as rows

Let us describ e now an imp ortant prop erty of these matrices for the construction of the resultant on an

irreducible variety X

Theorem Assume that

there exists a dense open subset X of the variety X such that the zeroset in X of the elements of

B B

x is empty X Z x

the conditions are satised

Then every minor D of size jB j of the matrix S is a multiple of the resultant Res f f

X n

Pro of We distinguish two cases Either S is always of rank jB j for any value of the parameters c and

any minor of size jB j is zero Then the theorem is obviously true

Or we may assume that S is generically of rank jB j Let Z b e the set of co ecient sp ecializations such

that f f have a common solution in X X Assume that there exists a nonvanishing maximal

minor of S Then S is surjective and any element of x is a p olynomial combination of the p olynomials

f f As these p olynomials have a ro ot X the elements x vanishes at which contradicts the

assumption

Therefore any maximal minor D of S is zero on Z thus it is zero on its Zariski closure which is

Z W for X is a dense subset of X The set of co ecients c Z is by denition the zero set of

are the co e Res f f Since the latter is irreducible it divides D in Zc where c

X i n ij ij j k

cients of f 2

In the examples that we will consider the condition will b e obviously true In the pro jective case

n n B

X P X will b e the ane space A and the set of monomial x will contain In the toric case X

n B n

K in X As the monomials x do not vanishes on K condition will b e the monomial image of

is satised

Remark This prop erty is interesting only in the case where the minors of size jB j are not all identically

D denote the degree in the co ecients of p olynomial f of a zero Assume this for a moment and let deg

Res f which is also the numb er generic of ro ots D deg nonzero maximal minor D of S Then deg

X c

f f

i i

D deg Res f and of Z f f f f If moreover jB j deg Res f we have deg

X c i i n X c

by cyclic p ermutation of the p olynomials and gcd computation we can recover the resultant see Mac

vdW CE

An imp ortant prop erty of this typ e of matrix from a computational p oint of view is its structure in the

sense of To eplitz and Hankel matrices see BP For general resultant matrices this kind of structure

was established by dening quasiTo eplitz and quasiHankel matrices in MPa MPb see also CKL

In particular Macaulay and toric resultant matrices exhibit quasilinear complexity for vector multiplica

tion EP MPb It has b een exploited in order to reduce by an order of magnitude the complexity of

solving some p olynomial systems MPa These matrices are also usually very sparse and this feature has

b een exploited in the algorithm prop osed in BMP for selecting the ro ots which maximize or minimize

a given criterion

Lots of questions in this direction need further investigations in order to understand deeply the structure

of these matrices For instance solving a To eplitz system can b e done in the univariate case in almost linear

time whereas for their generalization to the multivariate case only quasiquadratic algorithms are known

Moreover can we exploit b oth QuasiTo eplitz structure and sparsity

Macaulay matrices

Macaulays construction Mac of the resultant of n p olynomials f f of degree d d

n n

in the variables x x x pro ceeds as follows we give the nonhomogeneous version Let

n+1

B B

n+1

b e the set of all x d n et let x b e the set of all monomials in x of degree Let x

d d

n+1 n+1

B B B

n+1

x Among the remaining monomials in x x the monomials of x which are divisible by x

n n

d d

n n

Similarly for i n we dened by those which are divisible by x x let us denote by x

n n

d d

i+1 n+1

B B B B

i+1 n+1 i

which are divisible by x x x to b e the set of monomials of x x x induction x

i i

d d

B d B B B

i 2

n n+1 2 1

The set x x x x x x is denoted by x and is equal to

B b

1 n

fx b d g x x

i i

It has d d monomials Consider now the matrix S asso ciated to these subsets B It is a square matrix

n i

n+1

B B B B

n+1 1 2

of size the numb er of monomials of degree that is x x x for x x x

f f we rst In order to prove that its determinant is a nontrivial multiple of the resultant R

n P

prove that the conditions of theorem are true and then that the determinant of S is not zero Condition

is obviously true for x Condition is true according to section Now the determinant of S

for i n and f by we as a p olynomial in c c is not for when f is sp ecialized to x

ij i

obtain the identity matrix Thus according to theorem the determinant of this matrix is a nontrivial

multiple of the resultant Res f f

P n

This determinant is homogeneous of degree d d in the co ecient of f that is exactly the degree of

the resultant in f Therefore the resultant of these p olynomials can b e computed as the gcd of determinant

of such matrices obtained by a cyclic p ermutation of the p olynomials f f see remark

In fact FS Macaulay proved see Mac that the resultant is the ratio of det S by a determinant of

a submatrix of S

Example For the system

f c c x c x c x x

f c c x c x c x

the transposed Macaulay matrix has the form

c c c c

The i blo ck in the matrix corresp onds to the p olynomial f The size of these blo cks are for f for f

and for f We have B f x x x x g B f x x x x x g B f x x x x x x g

Here is a table giving the Bezout numb er d and the size of the Macaulay matrices for a linear form and

n p olynomials of degree d in n variables

nnd

These gures show the limit of the size of problems that can eectively b e treated by such metho ds The

next sections are devoted to metho ds which usually leads to smaller matrices

Newton matrices

The construction of Newton matrices is similar to the construction of Macaulay matrices except that it uses

more intricate geometry on the monomials in the f We briey sketch it b efore going into details

Let us x n Laurent p olynomials f f K x x with supp ort resp ectively in the p olytop es

Q Q Here is a short description of this algorithm

Construction of the Newton matrices

Sub divide the monomials of the Minkowski sum Q Q Q into cells mixed sub division

and faces of the other p olytop es of a vertex of some Q B Decomp ose each of these cell as a sum a

i i i

0 0 0

B B a

x to b e compared with the partition Q i i This gives a partition of x as an union of sets x

B B

in the previous section x of x into the union of the sets x

and construct the corresp onding by the p olynomial f For all these cells replace the monomial x

co ecient matrix of all these p olynomials

In order to get a matrix whose determinant is a nontrivial multiple of the resultant we will have to check

that the matrix is generically invertible

Let us describ e now more precisely the algorithm that guarantees most prop erties for the resultant

matrix The original version of CE was subsequently improved and generalized in CP as explained

at the end of the section A further generalization can b e found in Stu

Mixed sub division First we need to describ e how to sub divide the Minkowski sum of all input Newton

p olytop es Q Q Q R The basic construction extends that of mixed sub division describ ed

in section to an overconstrained system To apply the lifting technique select n linear lifting forms

l R R for i n Then dene the lifted Newton p olytop es

Q fp l p p Q g R i n

i i i i i i

and their Minkowski sum

b b b

Q Q Q R

The lower envelop e of Q pro jects bijectively onto Q by analogy to the case of n p olytop es discussed in

section The sub division of the lower envelop e into facets induces a sub division of Q into cells This is

known as a mixed subdivision and shall b e assumed xed in the course of the algorithm

The lifting functions l are chosen to b e suciently generic so that every p oint p Q can b e uniquely

written as a sum of Newton p olytop e p oints

p p p p Q i n

n i i

where the lifted images of the p add up to the unique p oint on the lower envelop e of Q which pro jects to

p The ab ove sum is called optimal b ecause the p minimize the aggregate lifting function l p over

i i i

all n tuples of p oints in R whose sum equals p Analogously each cell of the mixed sub division is

uniquely expressed as an optimal sum

F F R F is a face of Q i n

n i i

The lower envelop e facet pro jecting onto is the Minkowski sum of those faces in the Q corresp onding to

dim F and since dim n we deduce that at least one F is a F Genericity implies that dim

i i i

vertex Cells where exactly one summand face is a vertex are called mixed and in particular imixed if and

only if F is a vertex All other summand faces in a mixed cell must b e edges Recall that by denition

MV MV Q Q Q Q V i n

i i i n

imixed

The Minkowski sum contains all required information for the system as will b e made clear b elow

Partition of the monomials In order to remove the ambiguity for monomials in the b order of two cells

step of the algorithm we consider the set

B Q Z

where Q is a suciently small and generic vector The partition of B is the one induced by the mixed

sub division

Matrix construction The rows and columns of the toric resultant matrix S will b e indexed by B Every

pa p a

i i

0 0

is a vertex of where a x p oint p B is in an imixed cell Let us cho ose i maximal Then x x

B pa

Q Q By denition the column of hx i for p a x Q Note that f and p a Q

i i i i i i

0 0 0 0

i i

B pa

will b e called in the basis x The exp onent a x S indexed by p B is the co ecient vector of f

i i

0 0

the column content of p in S In summary we have asso ciated to every p oint in B the pro duct of some

p olynomial with a monomial thus dening a matrix row The ab ove arguments establish the fact that the

supp orts of all these pro ducts lie in B thus yielding the required closure property to guarantee that the

matrix is welldened and square Point set B and the corresp onding Minkowski sum Q are the smallest

p oint set and p olytop e resp ectively where this closure prop erty holds The role of the mixed sub division also

b ecomes obvious namely in order to sp ecify a unique sum of p oints dening each p oint in B

Nondegeneracy Every principal minor of matrix S including its determinant is nonzero when the

p olynomials have generic co ecients CE The pro of is based on the following technical lemma which

captures the geometric prop erties of the construction Sp ecialize S to a matrix S t where t is a new

where a suppf is the exp onent variable by sp ecializing every co ecient c in S to a p ower t

ij i ij

vector of the corresp onding monomial in f For every p E let pb b e the p oint on the lower envelop e of Q

lying directly ab ove p and let hpb express its n st co ordinate Consider any column of S t indexed

Then dene matrix S by multiplying the column of f by p B corresp onding to the p olynomial x

hpbl a

i ij

S t indexed by p by t

Lemma Assume the above notation and denote by S the entry of S with row index p and column

index q for p q E Then for al l nonzero elements S with p q deg S deg S

pq t pq t q q

According to theorem the determinant of S is divisible by the toric resultant Resf f

By construction the degree of det S in f is the numb er of p oints p B of the mixed cells where

F F F where dim F and dimF dim F This numb er is precisely

n n

the mixed volume of Q Q see section Thus the degree of det S in the co ecients of f equals

the degree of the toric resultant in the same co ecients The determinant degree in the co ecients of f for

i n is greater or equal to the resp ective degree of R According to remark we obtain the

resultant Res f by cyclic p ermutation of the f and gcd computation

X c i

Example The Newton matrix construction is illustrated for a system of p olynomials in unknowns

f c c xy c x y c x

f c y c x y c x y c x

f c c y c xy c x

The Newton p olytop es are shown in gure The mixed volumes are MV Q Q MV Q Q

a 22 a a 12 13 a32 a 33 a 21 a 23

a11 a 14 a 24 a31 a 34

Figure The Newton p olytop es and the exp onent vectors a

MV Q Q so the toric resultants total degree is Compare this with the Bzout numb ers

of the subsystems hence the classical resultant degree is We pick generic functions

l x y Lx L y l x y L x y l x y x Ly where L is a suciently large p ositive integer We

construct the Minkowski sum Q in three dimensions its lower envelop e is twodimensional and pro jects

Q Then we apply a p erturbation by vector bijectively onto Minkowski sum Q

i 3 22 22,33 y 32 11 33 33 y2 11,32 11,23 23 23 y 11 11 11 34 1

xx2 x3 x4

Figure The mixed sub division of Q where Each cell is lab eled with the indices

of the Newton p olytop e vertices that app ear in its optimal sum ij denotes vertex a

The mixed sub division of Q into dimensional cells and the indices of the Newton p olytop e vertices

in the optimal sums for each cell are shown in gure Matrix S the Newton matrix asso ciated to f

app ears b elow with rows and columns indexed by the integer p oints in E and has dimension S contains

by construction the minimum numb er of f rows namely The total numb er of rows is

Here is the transpose of S

1 0 2 0 0 1 1 1 2 1 3 1 0 2 1 2 2 2 3 2 4 2 1 3 2 3 3 3 4 3

3 2

1 0 c c 0 0 c c 0 0 0 0 0 0 0 0 0

11 14 12 13

7 6

2 0 c c 0 c c 0 0 0 0 0 0 0 0 0 0

31 34 32 33

7 6

0 1 0 0 c c 0 0 0 c c 0 0 0 0 0 0

11 14 12 13

7 6

1 1 0 0 0 c c 0 0 0 c c 0 0 0 0 0

11 14 12 13

7 6

2 1 c 0 c 0 c 0 0 0 c 0 0 0 0 0 0

24 21 23 22

7 6

3 1 0 c 0 c 0 c 0 0 0 c 0 0 0 0 0

24 21 23 22

7 6

0 2 0 0 c c 0 0 c c 0 0 0 0 0 0 0

31 34 32 33

7 6

1 2 0 0 0 c c 0 0 c c 0 0 0 0 0 0

31 34 32 33

7 6

2 2 0 0 0 0 0 0 0 0 c c 0 0 0 c c

11 14 12 13

7 6

3 2 0 0 0 0 c c 0 0 c c 0 0 0 0 0

31 34 32 33

7 6

4 2 0 0 0 0 0 c 0 0 c 0 c 0 0 0 c

24 21 23 22

7 6

1 3 0 0 0 0 0 0 0 c c 0 0 c c 0 0

31 34 32 33

7 6

2 3 0 0 0 c 0 0 c 0 c 0 0 0 c 0 0

24 21 23 22

7 6

5 4

3 3 0 0 0 0 0 0 0 0 c c 0 0 c c 0

31 34 32 33

4 3 0 0 0 0 0 0 0 0 0 c c 0 0 c c

31 34 32 33

The row and column indexed by can b e b oth removed thus factoring c out of the determinant

This is in fact the matrix obtained by a greedy variant of our algorithm prop osed by Canny and Peder

sen CP It constructs matrices whose size is typically smaller than and can never exceed that of the

original algorithm Moreover the greedy algorithm works on arbitrary supp orts thus removing the technical

requirement set b efore denition that the integer lattice they generate should b e fulldimensional

Namely the constructed matrix is square of dimension at most jB j Its determinant is a nontrivial multiple

det S of the toric resultant and its degrees in the co ecients of the given p olynomials f satisfy deg

Res for i n detS deg R and deg deg

f f f

i i 1

Incremental Newton matrices

In the previous section we describ es sets B and B which leads to a square nondegenerate resultant matrix

However e have no guarantee that its size is minimal A metho d prop osed in EC and implemented

in Emi consists to search incrementally subsets of the B and B which also yield square nongenerate

matrix This approach pro duces matrices whose dimension never exceeds that of the sub divisionbased

algorithms and which are typically signicantly smaller The exibility of the construction makes it suitable

for overconstrained systems On the other hand this is the reason that certain a priori prop erties of the

sub divisionbased construction are not guaranteed

We dene

n n

Q Q R and E Q Z for i n

i j i i

j j i

The algorithm shall restrict B to b e a subset of E just as in the case of the sub divisionbased algorithms

i i

This ensures that A B Q for all i The second concept used is the v distance of p oints in P Z

i i

n n

where P R is any convex p olytop e and v Q is a given vector

v distancep maxfs R p sv P g

This is the distance of p oint p from the p olytop e b oundary along direction v

The main issue is to cho ose the p oints of E that make up B The construction is incremental in the

i i

sense that successively larger candidate matrices are dened and tested for validity on whether they express

a nontrivial multiple of the toric resultant Supp osing that we are given a direction vector v we can partition

the p oints in every E with resp ect to their v distance At every step the algorithm adds to B all p oints in

i i

E whose v distance exceeds some b ound R for i n Incrementing the sets B is equivalent

i i

to decreasing until a valid matrix is found

For given sets B a rectangular matrix S is welldened and constitutes a useful candidate only if the

numb er of columns is at least as large as the numb er of rows If so the algorithm tests whether S has full

rank for generic co ecients in other words whether there exists a maximal minor D which is generically

nonzero The algorithm terminates if S has generically full rank and returns a nonsingular maximal square

submatrix This submatrix is a toric resultant matrix since its determinant D is a nontrivial multiple of R

Observe that the pro cess of deleting the extra columns do es not aect the validity of theorem

For D to b e a multiple of Res its degree must b e at least MV in the co ecients of f for i n

i i

Hence the initial sets B contain the MV p oints of largest v distance in E for i n The numb er

i i i

of p oints comprising each increment to the B s has b een studied in EP Large increments sp eed up the

construction but may miss the smallest p ossible matrix so some further tests may b e needed once a valid

matrix has b een found in order to decrease the matrix dimension It can b e shown that for v where is

the p erturbation vector in the sub divisionbased algorithm the incremental construction yields a matrix at

most as large Therefore if at some stage B E for i n then this v is rejected For arbitrary

i i

systems a random vector usually pro duces a smaller matrix But there is a class of systems for which a

deterministic vector guarantees the construction of an optimal matrix ie a matrix whose determinant

equals the resultant or equivalently a matrix whose dimension is minimum This class includes all systems

for which an optimal matrix of Sylvester typ e do es exist as discussed hereafter and in EC

There are two p otential b ottlenecks in this construction First enumerating all integer lattice p oints

in E for i n or rather an appropriate and suciently large subset of E Certain heuristics

i i

are prop osed and implemented in EC based on linear programming A more complete treatment of

the problem may b e based on enumerative geometric techniques GW Bar The second b ottleneck

is the fullrank test It can b e implemented in an incremental fashion based on an LU decomp osition of

rectangular matrices EC Extending CKL the results of EP establish a quasilinear complexity

for vector premultiplication which reduces to computing the sum of p olynomial pro ducts as seen ab ove

Applying standard results this yields quasiquadratic complexity for calculating the rank and the determinant

of a quasiTo eplitz matrix Wie GV BP Therefore the time complexity for matrix construction is

O n

O deg R t where t is the total numb er of rank tests during construction and vector v is assumed

O n

given This b ound b ecomes O deg R when the numb er of rows in the nal matrix is b ounded by a

constant multiple of deg R as is often the case in practice An interesting op en question is whether this

b ound can b ecome quasilinear

Compared to the matrix pro duced by a mixed sub division the incremental matrix has the following

features

There exist deterministic choices for vector v that yield optimal matrices for a sub class of multihomoge

neous systems as illustrated in example and in full detail in EC This sub class includes linear

systems and pairs of arbitrary p olynomials thus the algorithm of this section generalizes Sylvesters

construction

For suciently generic vector v this algorithm subsumes b oth the original sub divisionbased algorithm

and its greedy variant so it pro duces a matrix at most as large as those algorithms A brief explanation

is provided ab ove and a pro of in EC

Unlike the previous constructions where we could establish a closure prop erty like the one just in

section the heuristic nature of the incremental algorithm cannot provide the guarantee that every

principal submatrix is generically nonsingular EC To show an analogous closure prop erty here

would constitute a ma jor enhancement to this algorithm

If some set B is xed to its optimal size then we can apply the two alternatives in CE to recover

the actual resultant p olynomial

Example continued from section Figure shows Q in b old and randomly chosen vector

v The dierent subsets of E with resp ect to v distance are shown by the thinline p olygons

inside Q In fact the thin lines represent contours of xed v distance

Figure E subsets with dierent v distance b ounds and vector v

The nal set B includes all integer p oints in Q whose v distance is larger than or equal to

here is B with the v distances f g This v leads to a

nonsingular matrix S shown b elow with B cardinalities Recall that in general the algorithm constructs

a rectangular matrix from which it extracts a generically nonsingular maximal submatrix Here deleting the

last row denes the resultant submatrix The rst line b elow displays the integer p oints indexing the

columns Recall that the sub division and greedy algorithms give matrices of dimension and resp ectively

whereas the degrees of the toric and the classic resultant are and resp ectively Here is the transpose

of the constructed matrix

c c c c

Example Multihomogeneous systems

This section fo cuses on a sp ecial class of multihomogeneous p olynomial systems for which Sylvestertyp e

matrices provably exist for the toric resultant The incremental algorithm pro duces these matrices and

additionally nds rather compact matrices for arbitrary multihomogeneous systems

A homogeneous p olynomial is said to b e multihomogeneous if the set of variables can b e partitioned into r

subsets X X so that the p olynomial is homogeneous when considered as a p olynomial in each subset

This is sometimes called an r homogeneous p olynomial Supp ose that the numb er of individual variables in

X is l where one of them is the homogenizing variable then in our notation n l l If the

k k r

total degree in X is d then the p olynomial is said to b e of typ e l l d d There is a rich theory

k k r r

of multihomogeneous systems MS MSW namely systems where each p olynomial is multihomogeneous

and has the same supp ort If the degree of the ith p olynomial in X is d then the numb er of common

j ij

isolated solutions for the n p olynomials is b ounded by

n r r

Y X Y

d x in p olynomial x the co ecient of

ij j

i j j

Sturmfels and Zelevinsky SZ studied in particular the sub class of systems for which

l or d for k r

k k

They showed that every such system has a numb er of Sylvester typ e matrices for its toric resultant ie

matrices whose determinant is precisely the resultant Furthermore they conjectured that no other class

of systems has optimal matrices of Sylvester typ e ie where the matrix dimension is minimum and every

entry is either zero or an input co ecient It can b e proven that the optimal matrices whenever they exist

can b e constructed by the incremental algorithm for a deterministic choice of v

Theorem EC Consider a multihomogeneous system in the subclass specied above comprised of n

polynomials in n variables The variable set is partitioned in r subsets of l variables each for k r

so that al l polynomials have total degree d in the k th variable subset Using the above notation dene

vector v Q as a concatenation of r subvectors of cardinality l k r where al l coordinates in the

k th subvector are set to d d l l Then the incremental algorithm constructs an optimal

k k k k

j k

matrix for the input system

Exp erimental results show that for arbitrary multihomogeneous systems the same pro cedure for deter

mining v can b e applied to yield matrices of satisfactory size EC Table displays some such systems

the vector calculated by the pro cedure ab ove and the dimension of the matrix obtained by a random p er

turbation of this vector deg R indicates the total degree of the toric resultant and dim S is the dimension

of the constructed matrix

typ e vector v Q deg R dim S

Table Incremental matrix construction on multihomogeneous systems

Bzout matrices

In this section we recall some basic denitions from the theory of Bezoutians referring the reader to

CM EMa for further details This to ol generalizes the construction of E Bzout to the multivari

ate case In addition to the vector of variables x consider the vector y y y and write x x

(i) (i1)

px px

the discrete x y x x x y For a p olynomial q R dene p

n i

y x

i i

dierentiation of p For a sequence of n p olynomials f f f f R construct the following

p olynomial in x and y

f x f f

B C

det x y

f x f f

n n n n

where det denotes the determinant of the corresp onding matrix f f f f and and vary

in xed ranges This p olynomial of K x y is called the Bezoutian of f f f Note that it dep ends

on the order that we have imp osed on the variables Its expansion in the monomial basis of the form

x y y w x

denes a map

K R

The matrix of this map in the monomial basis is precisely the matrix of the co ecients

Denition For any sequence of n polynomials f f f f in n variables we denote by

B the matrix of the Bezoutian in the monomial basis We cal l it the Bezoutian matrix of f

Example continued from section The Bezoutian of

f c c x x c x x c x

f c x c x x c x x c x

f c c x c x x c x

is of the form

2 2 2 3 2 3 2 2 3 2

y + (c) x x y y y + (c) x x y y + (c) x y + (c) x x y y y + (c) x x y (c) x x

2 6 1 2 1 1 2 5 1 2 1 2 4 2 2 3 1 2 1 1 2 2 1 2 1 1 1 2

2 2 2 2 2 3 2 2

y + (c) x y y + (c) x y (c) x x y y + (c) x x y + (c) x y y + (c) x

2 12 2 1 1 11 2 1 10 1 2 1 2 9 1 2 2 8 2 1 1 7 2

2 2 2 2

y + (c) x x + (c) x y + (c) y y + (c) x y + (c) x x y + (c) x + (c) x x

2 18 1 2 19 2 1 17 1 1 16 2 1 14 1 2 1 15 2 13 1 2

+ (c) x + (c) y + (c) + (c) y

21 2 22 1 23 20 1

where (c) is a p olynomial in the parameters c = (c ) The Bezoutian matrix asso ciated to this p olynomial is the

i ij

following 5 5 matrix

3 2

[142] [114] + [411] [431] [134] [341] [143] [241] [421] [124] 0

7 6

[122] [224] + [423] [221] + [123] + [422] [322] [323]

7 6

[211] [113] [432] + [421] [342] [124] + [314] [311] [242] [413] + [214] + [132] [422] + [221] [123] [321]

7 6

[132] + [311] [343] + [433] [234] [314] [231] + [432] + [133] [342] [321] [324]

7 6

5 4

0 [422] [122] [222] [322]

where the symb ol [ij k ] stands for the pro duct c c c Its determinant equals c c c R 2

12 31

1(i1) 2(j 1) 3(k 1)

This matrix is usually of much smaller size than the resultant matrix as illustrated on this example

Let us denote by I the ideal generated by f f and by A the quotient algebra A R I An

imp ortant prop erty of these Bezoutians is given in the following prop osition see CM which is used in

the next theorem

set of equivalence classes mo dulo I

Prop osition Assume that the quotient A R I is a nite vector space of dimension d Then there

exist bases a b of R such that a a resp b b is a basis of A a I resp

i i d d d

in these bases is of the form b I and for any polynomial f R the matrix of B

d f f f

1 2 n+1

is the matrix of multiplication by f in the basis a a of A where M

d f

Let us describ e rst a direct application of this prop osition related to the Chow Form and based on the

are f for Z f f see AS Ste Mou fact that the eigenvalues of M

n f

Let us recall that the Chow form of f f is

u u u

n n

Z f f

2 n

where is the multiplicity of Z f f see HP It can b e shown that it is also the

where M is the matrix of multiplication by x in the quotient u M determinant det u I u M

x l n x d x

n 1

A R f f Therefore it is a homogeneous p olynomial in u u u with co ecients in K

n n

of A of total degree the dimension D

Z f f

2 n+1

If we are able to compute this Chow form or a multiple of it then by factorization of this p olynomial

K we can recover the linear factors in u over

u u u

n n

for Z I and thus the co ordinates of the ro ot Therefore computing this

n n

Chow form or a multiple of it can lead to an ecient way to nd the ro ots of the p olynomial equations An

algorithm for factoring such p olynomials has b een prop osed for instance in Car Given the Bezoutian

computing a multiple of this Chow form is a direct consequence of prop osition as explained b elow

Prop osition Any nonzero minor of maximal size of the Bezoutian matrix B

u u x u x f f

0 1 1 n n 2 n+1

u u u is divisible by the chow form C u

n n I

Z f f

2 n+1

The vanishing of the Chow form is a condition on the co ordinates u u u of an hyp erplane to

contain a ro ot of the system of equations f f It can b e seen as a sp ecial case of an

eliminant condition

We are going to show that the Bezoutian can b e used to obtain a nontrivial multiple of the general

resultant over a variety X when this is meaningful see section

We have dened Bezoutians for ane p olynomials but resultants are dened over pro jective varieties

n n

To work on an ane space we will consider a p olynomial map A X such that A X is dense

is well in X Then f f is a p olynomial in the variables x x x and the Bezoutian

i i n

f f

1 n+1

dened The next theorem shows that the resultant Res f f can b e recovered from the Bezoutian

X n

This result though a direct consequence of prop osition is new in the b est of our matrix B

f f

1 n+1

knowledge It generalizes the result of KSY where under a technical hyp othesis a multiple of the toric

resultant is computed

Theorem Assume that the conditions are satised and that A X is a polynomial map

is divisible such that its image is dense in X Then any maximal minor of the Bezoutian matrix B

f f

1 n+1

by the resultant Res f f

X n

Pro of According to the conditions the set of co ecients c c of f f such that Z f

ij n

n k k

n 1

As X A is a dense subset of X the set P f is nite is a dense subset of P

of co ecients c such that Z f f is nite and in X is also a dense subset Let us cho ose

ij n

generic co ecients in this dense subset for f f

Then the K vector space K x x f f is of nite dimension Let us denote by D the

n n g

generic dimension of this quotient For any f R we denote by r the generic rank of the Bezoutian matrix

are p olynomials in c which are not all identically zero and any The minors of size r of B B

f f f

1 1 n+1

minor of size r is identically zero

can b e decomp osed as in so According to prop osition for generic values of c the matrix B

that the rank of this matrix is

r ank L r ank M

f f

1 1

is the lower diagonal blo ck in the is the matrix of multiplication mo dulo f f and L where M

f f

1 1

decomp osition As for generic values of c the variety Z f f is empty condition

are the values of f at the is generically invertible the eigenvalues of M the multiplication matrix M

f f

1 1

R f f ro ots of f f that is of rank D dim

n n g K

Let us cho ose now f f such that their ro ots are in X and f has a common ro ot with f f

n n

D for f vanishes at one of the ro ots In this case Res f f Moreover we have r ank M

g X n

cannot exceed the generic rank Thus the matrix B of f f and by sp ecialization the rank of L

f f

1 1

is of rank r and all the r r minors vanish

g g g

As the set of systems f f such that Z f f X and f vanishes at one

n n

of these p oints is a dense subset of the resultant variety Z R es f f it implies that any

X n

maximal minor of the Bezoutian matrix vanishes on this resultant variety Consequently any maximal minor

of size r is divisible by the resultant which proves the theorem

An imp ortant asp ect of these construction is the size and the rank of the resulting Bezoutian matrix We

give here a b ound on the numb er of monomials when we x the degree d of the p olynomials f See CM

i i

for more details In this case we consider generic p olynomials of degree d and the b ounds given b elow have

to b e compared with those of Macaulay matrices or with the matrices asso ciated with simplices in the toric

case If d max d and n is the numb er of variables then the size of the Bezoutian matrix is b ounded by

i i

n n

e d

which is an order of magnitude less than the Macaulay matrix Here is a table of the size of the Bezoutian

matrix of f f for small values of the degree d d We give the size of the matrix its rank and

n n

we compare it with the Bzout b ound N which is a lower b ound on the generic rank of these matrices

d size rank N

d where can not exceed n A combinatorial result Hab shows that indeed the rank of B

f f

2 n+1

d max d For small value of n this is not to o far from d

in i

We end with some op en questions The maximal nonzero minors of the Bezoutian matrix are nontrivial

multiples of the resultant Can we obtain exactly the resultant by computing the gcd of these maximal

minorslike in Macaulay or Newton formulation Changing the variable order replacing the variables x by

see KS can b e helpful to remove some of the extraneous factors But even then the gcd of p owers x

all p ossible maximal minors may remain reducible as illustrated in the following example

f c c x c y

f c c x c y c x y

x y f c c x c y c

The Bezoutian matrix is a matrix whose determinant factors as

c c c c c c c c c c c c c c c c c c c c

Therefore an imp ortant problem is to describ e explicitly the factors that app ear in such a maximal minor

Dixon matrices

The two kinds of matrices that we have seen ie Bezoutian and Sylvester typ e matrices can b e mixed

together by cho osing some of the co ecients w x of the monomials y in the Bezoutian and some

multiples of the initial p olynomials f in order to build a square matrix

We called such matrices which combine blo cks of Sylvester and Bezout matrices Dixon matrices after

the work of AL Dixon see Dix who prop osed such matrix formulations for computing the resultant of

p olynomials over P This construction generalizes b oth Bzout and Sylvester construction of the resultant

We consider here two formulations of this typ e

The rst construction which yields to a smaller matrix than the Macaulay matrix from which the

resultant over X P can also b e constructed is as follows We consider the map

B B B

n+1 1

i K hx i i hx hx

q f w q q

i i n

where w is the constant co ecient in y of the Bezoutian of the homogeneous p olynomials f f

is the set of all monomials of degree where x d n and V is the set of all monomials of degree

j i

where d n Compared with the Macaulays formulation the matrix of is of size

which is less than the size of Macaulays matrix

The following theorem is essentially due to Macaulay

f f Prop osition Macaulay If is surjective then R es

n P

The converse is also true See also Jou Joua Joub for more details on dierent formulations of this

typ e and on pro jective resultant in several variables

f f implies that is not surjective or It is used in our case in the following way R es

n P

equivalently that all the maximal minors of the matrix of are divisible by the resultant

Prop osition The maximal minors of the matrix of are divisible by the resultant R f f

P n

of f f over P

This construction has also b een generalized to the case of toric varieties with some restriction on the supp ort

of the p olynomials in CD

In the second construction the numb er of columns from the Bezoutian matrix is greater and the numb er

of monomial multiples of the f from the Sylvester matrix is less This is precisely the construction prop osed

by AL Dixon for p olynomials in variables Dix that we generalize slightly The matrix that we

consider is the matrix of a map mixing the Bezoutian and the Sylvester approach of the form

B B B

n+1 1

i K K hx i i hx hx

k n

X X

w q f q q

i i i i n k

i i

where w are p olynomials which vanishes when the p olynomial f f have a common ro ot where k

i n

is the numb er of p olynomials w Here again to dene this map we have to sp ecify the set of monomials

B B B and the p olynomials w w

n k

The p olynomials w which were used in Dix are precisely the co ecients of the rst k monomials y

of smallest degree in the Bezoutian f f y w x

We assume for a moment that the p olynomials are of the same degree d We will take for x the set of

monomials of degree n d n u where u N will b e dened latter In order to to obtain a construction

which is symmetric in f f we will assume that the sets B are equal B B Let

n i n

l jB j jB j Now we adjust the parameters k and l in such a way that

the numb er k n l of columns of the matrix is the numb er jB j of rows

and the degree of the determinant of this matrix with resp ect to the co ecients of f is exactly d

that is the degree of the resultant in the co ecients of f

Thus we obtain the following constrains

ndu n

k n l k l d

for some u N or

n ndu ndu n

n d d

n n

N k N l

n n

for some value of u N

Example For n d and for the system of the form

i i i

1 2 3

f x x x c x x x for j

i i i i

1 2 3 4

i i i Ni i i

1 2 3 1 2 3

we obtain k l and the following matrix

x x x x x x x x

c c c f c

f c c c c

c c c x f c

x f c c

c x f c

x f c c

j j j

1 2 3

is the co ecient of y where w y y in the Bezoutian f f f f Thus each co ecient of w

j j j j j j

1 2 3 1 2 3

is a determinant of the matrix of co ecients of the quadratic forms f f The determinant of the

matrix is of degree with resp ect to each p olynomials This matrix has b een used to deduce a p olynomial

of degree in the direct kinematic problem of a parallel rob ot see Mou Moua Note that the

corresp onding Macaulay matrix is of size 2

It is not always p ossible to nd k and l satisfying these relations Dixon treated the case n where

solutions of these relations always exist and in the following table we give the rst other cases where we can

nd k l satisfying the relations In the nonempty entries of the table we have put the value of k l

and the size of the matrix constructed by this metho d

nnd 2 3 4 5 6 7 8 9

In these cases we have the prop erty

Prop osition The determinant of such a matrix when it exists is the resultant of the polynomials

f f over the projective space X P

An interesting challenge is to extend this construction to systems of equations f of degrees d not necessarily

i i

l l

s 1

of multihomogeneous equations AL P equal for the computation of resultants over X P

Dixon treated the case X P P and d d d n n m

Comparison b etween dierent matrices

This section fo cuses on some prop erties of the toric resultant matrices and compares them to the matrix

formulations of the classic resultant and to the Bzout typ e matrices We also conjecture the extension of

Macaulays exact rational expression to the context of toric resultants

The main characteristic of the Sylvestertyp e matrices is that the co ecients are either or the co ecients

of the input p olynomials This typ e includes the Macaulay and toric resultant or Newton matrices The

rows of these matrices corresp ond to multiples of the p olynomial f and the diculty for constructing these

matrices relies on the choice of these multiples In the case of toric resultants this is p erformed by geometric

considerations on the supp ort of the p olynomials

The sub division algorithms generalize the classical Macaulay construction in the sense that they pro duce

the same matrix on completely dense systems For instance if we take the following lifting and p erturbation

l L x L x x L x L x i n

i i i i i i i i n

l L x L x

n n n n

where L L L L

n n

we get the Macaulay matrix of f f Thus Macaulay matrices are a sp ecial case of Newton Matrices

The latter matrices are usually smaller but require more intricate computations on the p olytop es of the f

Macaulays impressive result is the derivation of the extraneous factor in the matrix determinant as a

minor of the matrix The extension of this formula to the case of the toric resultant matrix is a ma jor op en

question The natural way to generalize Macaulays result is by dening E E to b e the subset of these

p oints that do not lie in any imixed cell for any i f n g Let S denote the square submatrix

nm nm

of S that includes all entries whose row and column indices lie in E As noted ab ove S is generically

nonsingular Based on empirical results Canny and Emiris CE have stated the following conjecture

Conjecture There exist perturbation vector and lifting functions l l for which the determi

nant of matrix S divides exactly the determinant of Newton matrix S and hence the toric resultant of

the given polynomial system is R det S det S

The prop osed rational expression has the same degree as R in every p olynomial and furthermore equals

the resultant in the completely dense case by Macaulays result

Example continued from section Referring to the example of section we have

c c

and det S det S pro duces the correct toric resultant Hence the conjecture is veried here 2

The Sylvester typ e matrices admit a natural generalization to overconstrained systems f f m

n just by adding new blo cks S dep ending on the co ecients f f See for instance Laz

i n m

where following Macaulay Mac a rectangular matrix with jE j columns and more than jB j columns has

b een used See also CG GV Gri

Let us compare it with Bezoutian typ e matrices In practice the size of these Sylvestertyp e matrices is

usually larger than the size of the Bzouttyp e matrices In

The monomials in x resp y of x y also dep end on the p olytop e of the f In KSY for instance

f i

it is shown that the monomials in x resp y of the Bezoutian are in the Minkowski sum of pro jections on

some co ordinate hyp erplanes of the p olytop es generated by the p olynomials f

The entries of the Bezoutian are sums of determinants of the co ecients of the input p olynomials f for

the Bezoutian is obtained by expansion of a determinant of a matrix in p olynomials Thus this ob ject is

more dicult to compute compared with the Sylvester typ e matrices once the B are known

A numerical comparison in solving metho ds based on these matrices has b een initiated in EMb and

seems to give an advantage to the Bezoutian approach in terms of accuracy

The structure of the Bezoutian is more dicult to analyze but just as in the univariate case the reduction

of f f mo dulo the ideal f f leads to a matrix whose inverse is of Hankeltyp e

n n

Besides all these prop erties one ma jor advantage of Bezoutians is that they gives directly informations

on the isolated p oints of the variety see section and EM Moreover many interesting and p owerful

prop erties of the quotient algebra A R f f are connected to these matrices including duality

algebraic residues SS Kun EMa EMa real ro ot counting BCRS multiplicities Moub

Applications of resultant matrices

Monomial bases and multiplication maps

This section establishes certain facts ab out the co ordinate ring of the variety asso ciated to a wellconstrained

system Particularly useful are monomial bases since they index matrices that dene multiplication maps in

the corresp onding co ordinate ring Multiplication maps are directly obtained from resultant matrices given

by any of the formulations seen so far

The basic prop erty of all resultant matrices is that p ostmultiplication with certain column vectors ex

presses evaluation of the p olynomials whose co ecients have lled in the matrix rows For an arbitrary

K and S of Sylvester typ e

p q

p S

where p and q range over the p oints indexing rows and columns in S resp ectively and x f x denes the

contents of the row indexed by p The vector is indexed by the p oints q and contains the values

of column monomials x at The vector pro duct is indexed by p oints p and contains at the entry indexed

by p the value of x f x at For Macaulays matrix p oints p lie in the disjoint union of all sets B and

i i

q ranges over B following the notation of section In the toric context p oints p and q range over E in

the notation of section For the Bzouttyp e matrices we have to replace the multiples x f x by some

p olynomials w x which app ear in the expansion of f f

The p ostmultiplication prop erty has almost reduced the calculation of all common ro ots to computing

the kernel vectors of S In section we see that the problem may b e reduced to computing the eigenvalues

and right eigenvectors of a square matrix For now the p ostmultiplication prop erty is used in connection to

monomial bases and multiplication maps

Consider a wellconstrained system f f K x x If the ideal I generated by these p olynomials

K then its co ordinate ring A K x x I is a vector space corresp onds to a variety of zero dimension in

over K of nite dimension CLO Its dimension is the numb er of ro ots counted with multiplicity In the

toric case generically the vector spaces dimension is MV f f whereas in the pro jective case it is

deg f In the toric elimination context there is a stronger prop erty demonstrated in PS and also

proven directly from the sub divisionbased construction in ER

Prop osition Assume that Z f f is a nite set of simple roots Consider the integer lattice

Q after applying an innitesimal perturbation points lying in the mixed cel ls of the mixed subdivision of

as in section These points are in bijective correspondence with monomials in the n variables x and this

monomial set forms a vectorspace basis for K x x I

In establishing this prop osition the pro of of ER denes an overconstrained system by adding a generic

p olynomial f Then in appropriate mixed sub divisions of Q such as those of section the integer

p oints dening the monomial basis of K x x I are precisely those lying in the mixed cells

Let this p oint set b e denoted S E with cardinality MV f f The sets B and E n b partition

S into four blo cks S for i j where S and S are square of dimension jB j and jE n B j

resp ectively As discussed in section the sub divisionbased algorithm guarantees that submatrix S is

generically nonsingular so we can dene M S S S S of dimension jB j Then it is easy to

show that M denes a multiplication map in co ordinate ring A for p olynomial f This is an endomorphism

in A K x x I such that

g g f mo d I

In other words if p olynomial g A is represented by a row vector with resp ect to monomial basis B then

premultiplication of M by this vector yields another row vector expressing g f mo d I in the same basis

Information on a basis of the co ordinate ring can also b e recovered from the Bezoutian

Prop osition EMa Assume that Z f f is a nite set of points in A Then the set of

monomials in x respectively in y which appear in the Bezoutian of f f contains a basis of

coordinate ring K x x I or K xI

This also holds in the nongeneric case for any complete intersection

The advantage of obtaining a monomial basis in the nongeneric case provided the n p olynomials express

a complete intersection is imp ortant in practical applications of resultants We return to genericity issues

and their signicance in section

System solving by the uresultant

The main goal in system solving is to nd all common isolated ro ots of a wellconstrained system The

computation of the uresultant is a standard to ol for nding all isolated ro ots A related approach reduces

the problem to solving a single equation in one variable then lifting these solutions to the common ro ots of

the original system We restrict attention to zerodimensional systems dening a complete intersection Yet

the approach can b e extended to arbitrary systems including overconstrained ones through the techniques

of section This section continues the discussion ab ove and uses the same notation

Let f b e linear with all co ecients u b eing symb olic

n i

f u u x u x

n n n

so that f I generically We consider a sp ecialization of the co ecients of f f Whenever

n n

f vanishes at a common ro ot K of the original system then the resultant must

n n

vanish Therefore u u u divides R u u This is the classic approach based on the

n n n

uresultant for system solving vdW Mac Laz Can Ren The last three metho ds yield single

exp onential algorithms in the problem dimension for nding all isolated ro ots in the generic cases The

same is true with the toric resultant construction Instead of the Sylvestertyp e matrix for the construction

of the resultant it is p ossible to use the nonzero maximal minors of the Bezoutian of these p olynomials as

explained in prop osition In b oth cases we obtain a p olynomial in u which is divisible by the linear

factors u u u for Z f f

n n n

Let us describ e now two approaches which can b e sued at this p oint to recover the ro ots

K From the linear factors we deduce all First we compute explicitly the minor and factor it over

the co ordinates of the ro ots Of course factoring this p olynomial may give more linear factors than those

corresp onding to ro ots but this is not a severe limitation b ecause we still obtain a sup erset of all solutions

Algorithms for computing a numerical approximation of such factorization can b e found in Car for

instance

Another approach consists in computing the co ecient du of and the co ecients d u of x i

i i

n of this minor when we replace u by u v for some v K For this we dont necessarily need

i i i i

to compute the whole p olynomial which may b e huge Then we deduce a rational representation of the

ro ots d u x f u i n where f is a rational fraction in u See Macp Rou

i i i

The uresultant however is identically zero when the variety has p ositive dimension even if the p ositive

dimensional comp onent lies at innity This is pro jective innity for the classical context and toric innity for

the toric resultant as mentioned in section Then the techniques of section have to b e applied Note

however that the Bezoutian construction is still valid in the case where a p ositivedimensional comp onent

lies at innity or even in the ane part What we obtain here are the isolated ro ots of the variety see

System solving by eigenvectors

This section details the reduction of solving the initial nonlinear problem to an eigenproblem This is more

ecient than the ab ove metho d b ecause it do es not require factorization of large p olynomials

For this observe that the constant p olynomial f u has multiplication map u I where I is the

identity matrix of dimension dim A If now f u v x v x where v are random constants

K n n n i

M u I is of the form M by linearity prop erty the multiplication map of f denoted by M

f u n f

n+1 0 n+1

with v x v x Therefore when f then u is an eigenvalue of the matrix M

n n n

The obvious question then is what are the eigenvectors expressing Recall that p ostmultiplication by

K yields the values of the row a vector representing the evaluation of the column monomials at some

p olynomials at In practice only the constant co ecient of f is an indeterminate whereas all other

co ecients are sp ecialized to random values Then for any ro ot of f f all vectors of the form

where b ranges over B are eigenvectors of M based on expression The latter matrix has

numeric entries so the problem can b e solved by linear algebra metho ds Hence the eigenproblem oers a

necessary condition for the ro ots By exploiting the fact that the eigenvectors of interest exhibit a sp ecial

structure it is p ossible to recover all ro ot co ordinates AS MC Emi MPa

If we use Bezoutiantyp e matrices instead of Sylvestertyp e matrices we have to replace the eigenproblem

M u I v by the generalized eigenproblem B u B w When B is invertible these two

problems are equivalent and when B is not invertible compression techniques can b e used in order to

reduce the generalized eigenproblem to a nonsingular one see Mou

We have reduced ro ot nding to a problem in linear algebra by adding the uform to the given well

constrained system An alternative is to hide one of the n variables in the co ecient eld This pro duces an

overconstrained system without increasing the problem dimension Our exp erience with systems in rob otics

and vision suggests that this is preferable in many practical situations Emi Formally we consider the

given p olynomials as

x x f f K x x x

n n n

Variable x is chosen so that all ro ots are separated by pro jection on x if p ossible Otherwise we have to

n n

deal with the case of multiple ro ots The construction of S is as b efore and any algorithm can b e used There

is a generating set of the co ordinate ring dened and made up of monomials Moreover a multiplication

map of largerthanoptimal dimension is in general obtained and a nontrivial multiple of the uresultant can

b e computed S is dened by eliminating the largest p ossible numb er of columns in S which are constant

with resp ect to x Then the problem is reduced to an eigendecomp osition of S

Row and column p ermutations do not aect the matrix prop erties so we apply them to obtain a maximal

S Gaussian elimination of the leftmost set of columns is now p ossible essentially giving S x S x

n n

S x In contrast to the previous approach the matrix may b e nonlinear in the hidden variable S x S

n n

so S x is matrix p olynomial A x A x A for some d If A is numerically nonsingular

n d n d

d d

we reduce the equation Ix A A x A A v to the following eigenproblem

n n

d d

w x w

A A A A A A A A

d d

d d d d

where w v x v x v If A is numerically singular we may change variable x to t y t t y t

n d n

for random t t This rankbalancing improves the numeric rank of A If the latter is still nonsingular

we have to consider a generalized eigenproblem on the following matrix p encil

I I

y C y

I I

A A A A

d d

For every eigenvalue with asso ciated right eigenvector w v v of C y we have v v for

d i

i d Moreover Ay has the same eigenvalue and has right eigenvector v These are all standard

op erations in numerical linear algebra GV This is also valid for Bezoutiantyp e matrices if we replace

eigenvector problems by generalized eigenvector problems

Indeed one can apply quite general metho ds to a numb er of dierent resultant matrices for solving

p olynomial systems by numerical linear algebra These numerical matrix manipulations manage to obtain

smaller matrices that contain all relevant information through a Schur factorization Jordan decomp osition

or singular value decomp osition resp ectively MD MS CGT or computing maximal nonsingular

minors KSY MZW CM

Several implementations of resultant matrices use these matrix manipulations in order to solve systems

of arbitrary p olynomial equations We mention those by Mano cha and coworkers KM and by Canny and

Emiris Emi A similar approach generalizing Lazards metho d Laz and implemented in Matlab is

prop osed by Corless Gianni and Trager CGT The second author is developing a program for system

solving based on the Bzout matrix MPa and so are Kapur and Saxena KSY A C library linking

some linear algebra packages such as Lapackor umfpack called ALP is currently under development in

order to gather all these dierent techniques see httpwwwinriafrsagalogicielsALP

Genericity issues

We have seen that resultants provide exact conditions for the existence of ro ots only in pro jective space

or in some toric variety Otherwise they provide only necessary conditions The computation of resultant

matrices may b e fruitless in the case of sp ecic co ecient sp ecializations for instance when the underlying

system has p ositivedimensional comp onents in the pro jective variety X For instance a p ositivedimensional

comp onent at innity causes the resultant p olynomial to vanish identically In these degenerate cases the

constructed matrices may b e identically singular thus oering no indication of the vanishing of the resultant

Recent work fo cuses on the degenerate cases and adapts resultant theory so that it applies to arbitrary

inputs One of the goals is to extract useful information on the isolated ro ots even in the presence of

p ositivedimensional comp onents by generalizing the classic resultant and the toric resultant The last part

of the section considers linear algebra manipulations that may prove more ecient in handling degenerate

cases

In the context of the classical theory Canny analyzed the generalized characteristic p olynomial for

computing the resultant over general algebraically closed elds Ren Can Its name is due to the fact

that it generalizes the characteristic p olynomial of linear systems See also Chi Gri It provides a

pro jection op erator that is not identically zero in the presence of p ositivedimensional comp onents at innity

or elsewhere The input p olynomial system is p erturb ed by p olynomials dened on the new supp orts and

multiplied by The resultant of the p erturb ed system is an p olynomial and its constant term is identically

zero if and only if the systems variety has p ositive dimension in an appropriate variety Its nonzero co ecient

of lowest degree generalizes the resultant and in particular the uresultant in the case that a uform is the

rst p olynomial This op erator generalizes the uresultant and oers a necessary condition for the existence

of isolated ro ots Although the construction was prop osed for Macaulays matrix it can also b e coupled

with Lazards matrix Laz The generalized characteristic p olynomial has b een recently adapted to the

toric context by Ro jas Ro ja

Let us mention here that these p erturbation techniques are not required for Bezoutian matrices Indeed

in the case of any ane complete intersection whatever the situation at innity is the maximal minors of

these matrices yield a multiple of the uresultant or Chow form see prop osition But even when the

variety has a p ositivedimensional comp onent in the ane part these minors give a nontrivial multiple of

the Chow form of the isolated roots see EM

These metho ds dene a p erturb ed determinant in terms of some parameter Recovering the trailing term

of this determinant typically requires some exact computation that increases the practical complexity of

the problem On the contrary the approach Mou avoids degeneracies purely by matrix op erations and

applies for dierent kind of formulations It works directly with the degenerate resultant matrices and

more precisely with p encils of matrices of the form M z M asso ciated to these resultant matrices The

i i

socalled Kronecker decomp osition of such p encils yields a left and a right singular part and a regular part

which can b e used to solve the system This metho d has the advantage to b e practical even with approximate

co ecients stable algorithms b eing available for computing this regular part eg DK

Conclusion

Sparse elimination theory is a comparatively recent algebraic approach studied in the past two decades and

dealing with p olynomials describ ed by their monomial supp orts This leads to more ecient algorithms in

practice and calls for several combinatorial and geometric techniques Bzout and Dixon matrices provide

more compact conditions and may prove to b e numerically more stable In addition to a general overview

here we have established a new result concerning the relevance of minors in the Bzout matrix This survey

has reviewed the state of the art in constructing resultant matrices the ma jor step in reducing system

solving to a problem in linear algebra as well as for computing the toric resultant p olynomial We have also

describ ed several metho ds for solving arbitrary systems of p olynomial equations including the degenerate

cases We have also p ointed out main op en issues in this domain which is exp ected to b e equally active in

the years to come

Acknowledgments

The rst author thanks Maurice Ro jas for eagerly providing preprints and explanations of his work Both

authors partially supp orted by Europ ean ESPRIT pro ject FRISCO LTR

References

AK LA Aizenb erg and A M Kytmanov Multdimensional analogues of newtons formulas for systems of

nonlinear algebraic equations and some of their applications Trans from Sib Mat Zhurnal

AS W Auzinger and HJ Stetter An elimination algorithm for the computation of all zeros of a system of

multivariate p olynomial equations In Proc Intern Conf on Numerical Math volume of Int Series

of Numerical Math pages Birkhuser Verlag

B E Bzout Thorie Gnrale des quations Algbriques Paris

Bar AI Barvinok A p olynomial time algorithm for counting integral p oints in p olyhedra when the dimension

is xed In Proceedings of the th Annual Symposium on Foundations of Comptuer Science pages

Palo Alto CA Novemb er IEEE Computer So ciety Press

BCRS E Becker JP Cardinal MF Roy and Z Szafraniec Multivariate Bezoutians Kronecker symb ol and

EisenbudLevin formula In L Gonzles and T Recio editors Algorithms in Algebraic Geometry and

Applications volume of Prog in Math pages Birkuser Basel

Ber DN Bernstein The numb er of ro ots of a system of equations Funct Anal and Appl

BGVY CA Berenstein R Gay A Vidras and A Yger Residue Currents and Bezout Identities volume of

Prog in Math Birkhuser

BGW C Ba ja j T Garrity and J Warren On the applications of multiequational resultants Technical Rep ort

Purdue Univ

BMB LM Balb es SW Mascarella and DB Boyd A p ersp ective of mo dern metho ds in computeraided drug

design In KB Lipkowitz and DB Boyd editors Reviews in Computational Chemistry volume pages

VCH Publishers New York

BMP D Bondyfalat B Mourrain and VY Pan Controlled iterative metho ds for solving p olynomial systems

In ISSAC ACM Press

BP D Bini and V Pan Polynomial and matrix computations Vol Fundamental Algorithms Birkhuser

Boston

BPR S Basu R Pollack and MF Roy On the combinatorial and algebraic complexity of quantier elimi

nation In Proc IEEE Symp Foundations of Computer Science Sante Fe New Mexico

BS LJ Billera and B Sturmfels Fib er p olytop es Annals of Math

BY C A Berenstein and A Yger Eective Bezout identities in Q [z z ] Acta Math

1 n

BZ YD Burago and VA Zalgaller Geometric Inequalities Grundlehren der mathematischen Wis

senschaften Springer Berlin

Can JF Canny The Complexity of Robot Motion Planning MIT Press Cambridge Mass

Can J Canny Generalised characteristic p olynomials J Symbolic Computation

Can J Canny Improved algorithms for sign determination and existential quantier elimination The Com

puter Journal

Car Carstensen C On Graus metho d for simultaneous factorization of p olynomials SIAM J Numer Anal

Car JP Cardinal Dualit et algorithmes itratifs pour la rsolution de systmes polynomiaux PhD thesis

Univ de Rennes

CD E Cattani and A Dickenstein A global view of residues in the torus J of Pure and Applied Algebra

CE J Canny and I Emiris An ecient algorithm for the sparse mixed resultant In G Cohen T Mora

and O Moreno editors Proc Intern Symp on Applied Algebra Algebraic Algor and ErrorCorr Codes

Lect Notes in Comp Science pages Puerto Rico Springer

CE JF Canny and IZ Emiris A sub divisionbased algorithm for the sparse resultant J ACM

Submitted

CG AL Chistov and DY Grigoryev Complexity of Quantier Elimination in the Theory of Algebraical ly

Closed Fileds Lect Notes Comp Sci Springer New York

CGT RM Corless PM Gianni and BM Trager A reordered Schur factorization metho d for zerodimensional

p olynomial systems with multiple ro ots In Proc ACM Intern Symp on Symbolic and Algebraic Com

putation pages

Chi AL Chistov Algorithm of p olynomial complexity for factoring p olynomials and nding the comp onents

of varieties in sub exp onential time J Sov Math

CKL JF Canny E Kaltofen and Y Lakshman Solving systems of nonlinear p olynomial equations faster

In Proc ACM Intern Symp on Symbolic and Algebraic Computation pages

CLO D Cox J Little and D OShea Ideals Varieties and Algorithms Undergraduate Texts in Mathematics

Springer New York

CM JP Cardinal and B Mourrain Algebraic approach of residues and applications In J Renegar M Shub

and S Smale editors Proc AMSSIAM Summer Seminar on Math of Numerical Analysis Park City

Utah volume of Lectures in Applied Math pages Am Math So c Press

Cox D Cox The homogeneous co ordinate ring of a toric variety J Algebraic Geom

CP J Canny and P Pedersen An algorithm for the Newton resultant Technical Rep ort Comp Science

Dept Cornell University

CR J Canny and JM Ro jas An optimal condition for determining the exact numb er of ro ots of a p olynomial

system In Proc ACM Intern Symp on Symbolic and Algebraic Computation pages Bonn July

Dan VI Danilov The geometry of toric varieties Russian Math Surveys

Dix AL Dixon The eliminant of three quantics in two indep endent variables Proc of Lond Math Society

DK J Demmel and B Kgstrm The Generalized Schur Decomp osition of an Arbitrary Pencil A B

Robust Software with Error Bounds and Applications II I ACM Trans on Math Software

DL DKapur and YN Lakshman Elimination metho ds an intro ducton In B Donald D Kapur and

J Mundy editors Symbolic and Numerical Computation for Artitial Intel lingence pages Aca

demic Press New York

EC IZ Emiris and JF Canny Ecient incremental algorithms for the sparse resultant and the mixed

volume J Symbolic Computation August

EMa M Elkadi and B Mourrain Appro che Eective des Rsidus Algbriques Rapp ort de Recherche

INRIA

EMb I Z Emiris and B Mourrain Polynomial system solving the case of a 6atom molecule Rapp ort de

Recherche INRIA Dec

EM M Elkadi and B Mourrain Some applications of b ezoutians in eective algebraic geometry Rapp ort de

recherche INRIA

Emi IZ Emiris On the complexity of sparse elimination J Complexity

Emi IZ Emiris A general solver based on sparse resultants Numerical issues and kinematic applications

Technical Rep ort INRIA SophiaAntip olis France

EP IZ Emiris and VY Pan The structure of sparse resultant matrices In Proc ACM Intern Symp on

Symbolic and Algebraic Computation pages Maui Hawaii July

ER IZ Emiris and A Rege Monomial bases and p olynomial system solving In Proc ACM Intern Symp

on Symbolic and Algebraic Computation pages Oxford July

FGS N Fitchas M Giusti and M Smietanski Sur la complexit du thorme des zros In J Gudatt editor

Proc of the second Int Conf on Approximation and Optimization Peter Lang Verlag La Habana

Ful W Fulton Introduction to Toric Varieties Numb er in Annals of Mathematics Princeton University

Press Princeton

GKZ IM Gelfand MM Kapranov and AV Zelevinsky Discriminants Resultants and Multidimensional

Determinants Birkhuser BostonBaselBerlin

Gri DY Grigoryev Factorization of p olynomials over nite eld and the solution of systsems of algebraic

equations J Sov Math

Gri DY Grigoryev The complexity of deciding Tarski algebra J Symbolic Computation

Gr B Grnbaum Convex Polytopes WileyInterscience New York

GV DY Grigoryev and NN Vorob jov Solving systems of p olynomial inequalities in sub exp onential time

J Symbolic Computation Sp ecial Issue on Decision Algorithms for the Theory of Real Closed Fields

GV GH Golub and CF Van Loan Matrix Computations The Johns Hopkins University Press Baltimore

Maryland rd edition

GW PM Grub er and JM Wills editors Handbook for Convex Geometry North Holland Amsterdam

Hab L Habsieger Sur un problme combinatoire Communication p ersonnelle

Har J Harris Algebraic Geometry a srt course volume of Graduate Texts in Math Springer

Hof CM Homann Geometric and Solid Modeling Morgan Kaufmann

HP W Ho dge and D Pedo e Methods of algebraic geometry Cambridge University Press

HS B Hub er and B Sturmfels A p olyhedral metho d for solving sparse p olynomial systems Math Comp

HS B Hub er and B Sturmfels Bernsteins theorem in ane space Discr and Computational Geometry

March

Jou JP Jouanolou Le formalisme du rsultant Adv in Math

Joua JP Jouanolou Formes dinertie et Rsultants Un formulaire Prpublication de l IRMA Strasb ourg

Joub JP Jouanolou Rsultant anisotrop e complments et appplications Prpublication de l IRMA Stras

b ourg

Kho AG Khovanskii Newton p olyhedra and the genus of complete intersections Funktsionalnyi Analiz i

Ego Prilozheniya JanMar

KM S Krishnan and D Mano cha Numericsymb olic algorithms for evaluating onedimensional algebraic sets

In ISSAC pages Montreal Canada ACM Press

Knu DE Knuth The Art of Computer Programming Seminumerical Algorithms volume AddisonWesley

Reading Massachusetts

KS D Kapur and T Saxena Extraneous Factors in the Dixon Resultant Formulation In WW Kchlin

editor ISSAC pages ACM Press

KSY D Kapur T Saxena and L Yang Algebraic and geometric reasoning using Dixon resultants In

ISSAC Oxford England

KSY D Kapur T Saxena and Lu Yang Sparsity Considerations in Dixon Resultants In STOC pages

ACM Press

KSZ MM Kapranov B Sturmfels and AV Zelevinsky Chow p olytop es and general resultants Duke Math

Kun E Kunz Khkler dierentials Advanced lectures in Mathematics Friedr Vieweg and Sohn

Kus AG Kushnirenko Newton p olytop es and the Bzout theorem Funktsionalnyi Analiz i Ego Prilozheniya

JulSep

Las A Lascoux La rsultante de deux p olynmes In Sminaire MP Mal liavin volume of Lec Notes

in Math pages

Laz D Lazard Rsolution des systmes dquations algbriques Theo Comp Science

LM T Lickteig and K Meer A note on testing the resultant J of Complexity

LWW TY Li T Wang and X Wang Random pro duct homotopy with minimal BKK b ound In J Renegar

M Shub and S Smale editors The Mathematics of Numerical Analysis volume of Lectures in Applied

Math AMS

Mac FS Macaulay Some formulae in elimination Proc London Math Soc

Mac FS Macaulay On the Resolution of a given Mo dular System into Primary systems including some

Prop erties of Hilb ert Numb ers Proc London Math Soc pages

Mac FS Macaulay The Algebraic Theory of Modular Systems Cambridge Univ Press

MC D Mano cha and J Canny Multip olynomial resultants and linear algebra In Proc ACM Intern Symp

on Symbolic and Algebraic Computation pages

MD D Mano cha and J Demmel Algorithms for intersecting parametric and algebraic curves I I Multiple

intersections Graphical Models and Image Proc

Mou B Mourrain The generic p ositions of a parallel rob ot In M Bronstein editor ISSAC ACM press

pages Kiev Ukraine July

Moua B Mourrain Enumeration problems in Geometry Rob otics and Vision In L Gonzles and T Recio

editors Algorithms in Algebraic Geometry and Applications volume of Prog in Math pages

Birkuser Basel

Moub B Mourrain Isolated p oints duality and residues J of Pure and Applied Algebra

Sp ecial issue for the pro c of the th Int Symp on Eective Metho ds in Algebraic Geometry

MEGA

Mou B Mourrain Computing isolated p olynomial ro ots by matrix metho ds J Symbolic Computation Special

Issue on SymbolicNumeric Algebra for Polynomials

MPa B Mourrain and V Y Pan Multidimensional structured matrices and p olynomial systems Calcolo

Special Issue workshop on Toeplitz matrices Structure Algorithms and Applications

MPb B Mourrain and V Y Pan Solving sp ecial p olynomial systems by using structured matrices and algebraic

residues In F Cucker and M Shub editors Proc of the workshop on Foundations of Computational

Mathematics Rio de Janeiro pages Springer

MPa B Mourrain and VY Pan Asymptotic acceleration of solving multivariate p olynomial systems of equa

tions In STOC ACM Press

MPb B Mourrain and VY Pan Multivariate p olynomials duality and structured matrices Rapp ort de

recherche INRIA

MS AP Morgan and AJ Sommese A homotopy for solving general p olynomial systems that resp ects m

homogeneous structures Appl Math Comput

MS HM Mller and HJ Stetter Multivariate p olynomial equations with multiple zeros solved by matrix

eigenproblems Numer Math

MSW AP Morgan AJ Sommese and CW Wampler A pro ductdecomp osition b ound for Bzout numb ers

SIAM J Numerical Analysis

Mui T Muir History of determinants Dover reprints volumes

MZW D Mano cha Y Zhu and W Wright Conformational analysis of molecular chains using nanokinematics

Computer Applications of Biological Sciences

Ped P Pedersen AMSIMSSIAM Summer Conference on Continuous Algorithms and Complexity

Mt Holyoke Mass July

PS P S Pedersen and B Sturmfels Pro duct formulas for resultants and Chow forms Math Zeitschrift

PS P Pedersen and B Sturmfels Mixed monomial bases In L GonzalezVega and T Recio editors Eective

Methods in Algebraic Geometry volume of Progress in Mathematics pages Boston

Birkhuser Pro c MEGA Santander Spain

Ren J Renegar On the worstcase arithmetic complexity of approximating zeros of system of p olynomials

SIAM J of Computing

Ren J Renegar On the computational complexity and geometry of the rst order theory of reals I I I I I I

J Symbolic Computation

Ro j JM Ro jas A convex geometric approach to counting the ro ots of a p olynomial system Theor Comp

Science

Ro ja JM Ro jas Toric generalized characteristic p olynomials Technical Rep ort MSRI Berkeley

Submitted for publication

Ro jb JM Ro jas Toric intersection theory for ane ro ot counting J Pure and Applied Algebra To

app ear

Ro jc JM Ro jas Toric laminations sparse generalized characteristic p olynomials and a renement of Hilb erts

tenth problem In F Cucker and M Shub editors Proc Workshop on Foundations of Computational

Mathematics pages Berlin Springer

Rou F Rouillier Algorithmes ecaces pour ltude des zros rels des systmes polynomiaux PhD thesis

Universit de Rennes

RR M Raghavan and B Roth Solving p olynomial systems for the kinematics analysis and synthesis of

mechanisms and rob ot manipulators Trans ASME Special th Annivers Design Issue

June

Sch R Schneider Convex Bodies The BrunnMinkowski Theory Cambridge University Press Cambridge

Sha IR Shafarevitch Basic Algebraic Geometry Springer Verlag

SS G Scheja and U Storch b er Spurfunktionen b ei vollstndigen Durschnitten Journal Reine Angew

Mathematik

SS J Sabia and P Solerno Bounds for traces in complete intersections and degree in the nullstellenstaz

AAECC

Ste H J Stetter Eigenproblems are at the Heart of Polynomial System Solving SIGSAM Bul letin

Stu B Sturmfels On the Newton p olytop e of the resultant J of Algebr Combinatorics

Syl JJ Sylvester On a theory of syzygetic relations of two rational integral functions comprising an applica

tion to the theory of Sturms functions and that of the greatest algebraic common measure Philosophical

Trans

SZ B Sturmfels and A Zelevinsky Multigraded resultants of Sylvester typ e J of Algebra

vdW BL van der Waerden Modern Algebra F Ungar Publishing Co New York rd edition

VGC J Verschelde K Gatermann and R Co ols Mixed volume computation by dynamic lifting applied to

p olynomial system solving Discr and Comput Geometry

VVC J Verschelde P Verlinden and R Co ols Homotopies exploiting Newton p olytop es for solving sparse

p olynomial systems SIAM J Numerical Analysis

Wie DH Wiedemann Solving sparse linear equations over nite elds IEEE Trans Inf Theory

Zip R Zipp el Eective Polynomial Computation Kluwer Academic Publishers Boston