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Home , Degree of a polynomial

Synthesis of Parallel Algorithms

Chapter June



Parallel

Computation

Doug Ierardi

Department of Computer Science

University of Southern California

Los Angeles California

Dexter Kozen

Department of Computer Science

Cornel l University

Ithaca New York

A resultant is a purely algebraic criterion for determining whether a nite

collection of p olynomials have a common zero It has b een shown to b e a useful

to ol in the design of ecient parallel and sequential algorithms in symb olic

algebra computational geometry computational numb er theory and rob otics

We b egin with a brief history of and a discussion of some of

their imp ortant applications Next we review some of the mathematical back

ground in commutative algebra that will b e used in subsequent sections The

Nullstellensatz of Hilb ert is presented in b oth its strong and weak forms We

also discuss briey the necessary background on graded algebras and dene

ane and pro jective spaces over arbitrary elds We next present a detailed

account of the resultant of a pair of univariate p olynomials and present e

cient parallel algorithms for its computation The theory of subresultants is

develop ed in detail and the computation of p olynomial remainder sequences is

derived A resultant system for several univariate p olynomials and algorithms

for the gcd of several p olynomials are given Finallywedevelop the theory

of multivariate resultants as a natural extension of the univariate case Here

we treat b oth classical results on the pro jective homogeneous case as well

as more recent results on the ane inhomogeneous case The uresultant of

a set of multivariate p olynomials is dened and a parallel algorithm is pre

sented We discuss the computation of generalizedcharacteristic

and relate them to the decision problem for the theories of real closed and

algebraically closed elds

Intro duction

The sub ject of this chapter is the computation of resultants A resul

tant is a purely algebraic criterion for determining whether a nite collection

of p olynomial has a common solution expressed in terms of the

co ecients of these p olynomials The investigation of such criteria b elongs

historically to the branch of known as elimination theorythe

goal of whichwas to solve systems of p olynomial equations by successive elim

ination of variables Fundamental asp ects of this pro ject were develop ed by

Hermann Hilb ert Kronecker Lasker Macaulay and No ether at the turn of

this century marking the b eginning of a fusion of algebra and geometry which

later found fuller expression in the development of algebraic geometry

 Chapter Parallel Resultant Computation

Much of the fundamental work in elimination theory was pursued at a

time when constructive metho ds in mathematics prevailed In fact the osten

sible goal of the theorysolving systems of p olynomial equationshad such

obvious practical signicance that the eciency of algorithms was already a

concern Macaulay expressed this p oint of view in his tract The Alge

braic Theory of Modular Systems when he wrote that the currentbodyof

knowledge in eliminationtheory

might b e regarded as in some measure complete if it were admit

ted that a problem is nished with when its solution has b een

reduced to a nite numb er of feasible op erations If however the

op erations are to o numerous or to o involved to b e carried out in

practice the solution is only a theoretical one and its imp ortance

then lies not in itself but in the theorems with whichitisasso

ciated and to which it leads Such a theoretical solution must b e

regarded primarily as a preliminary and not the nal stage in the

consideration of the problem

The study of algorithms in elimination theory has not yet reached its nal

stage provably optimal algorithms are still lacking Nevertheless contemp o

rary ideas in algorithm design and complexity are continually b eing brought

to b ear and the most ecientsequential and parallel algorithms for these

problems have b een discovered during the last decade

The theory of resultants rests on the well known Nullstellensatz of

Hilb ert which relates the algebra of the of p olynomials over indeter

minates x x with co ecients in an algebraically closed eld k denoted

n

n

k x x and the geometry of the space k of ntuples of elements of k

n

For example over the complex numb ers C the Nullstellensatz asserts that m

p olynomials f f with complex co ecients have no common solutions

m

n

in C exactly when there are m additional p olynomials g g such that

m

f g f g f g

m m

The foundation of elimination theory lies in the fact that the existence or

nonexistence of these g s can b e determined solely by examining the co e

i

cients of the f s and that an algebraic criterion can b e constructed uniformly

i

once the degrees of these p olynomials are sp ecied From this observation the

notion of a resultant arosea p olynomial in the co ecients of the given p oly

nomials whichvanishes exactly when a common solution exists

Intro duction 

Stated in this way it might seem that the resultant yields no more than

a decision pro cedure for the existence of solutions but the fact that it provides

a purely algebraic criterion extends its usefulness signicantly Resultants can

b e used in constructing solutions to systems of equations b oth symb olically

and numerically byapproximation They have b een employed successfully

in the design of ecient parallel and sequential algorithms in symb olic algebra

computational geometry and computational number theory and havefound

imp ortant practical applications in solid mo deling and rob otics

Outline of this Chapter

This chapter presents parallel algorithms to compute the resultants of b oth

univariate and multivariate p olynomials

Webegininx with a review of some of the mathematical back

ground in commutative algebra that will b e required including the necessary

facts regarding graded algebras and ane and pro jective spaces over arbi

trary elds The Nullstellensatz of Hilb ert is presented in b oth its strong and

weak forms

In x we give a detailed account of the construction of the resul

tant of a pair of univariate p olynomials The treatment is also extended to

deal also with several p olynomials in a single In exploring prop erties

of these calculations the theory of subresultants is develop ed in detail and

an ecient parallel algorithm for the computation of remainder

sequences is derived in a natural wayWe discuss the applications of subresul

tants in parallel greatest common divisor gcd algorithms and in computing

the extended Euclidean scheme These algorithms haveplayed a ma jor role

in the recentdevelopment of parallel metho ds in real geometryFor example

the algorithm of BenOr Kozen and Reif whichgives an ecient parallel

decision pro cedure for questions in the theory of real closed elds employs

avariety of parallel resultantbased techniques the extension and corrected

analysis by Renegar makes essential use of multivariate resultants pre

sented in x b elow Although a complete discussion of these applications

is b eyond the scop e of this chapter our presentation should b e sucientto

enable the interested reader to pursue more advanced topics in the literature

References to such applications are provided at the end of this chapter

In x the theory of multivariate resultants is develop ed as a natural

extension of the univariate case Here we treat b oth classical results on the

 Chapter Parallel Resultant Computation

pro jective homogeneous case as well as more recent results on the ane

inhomogeneous case In this section our presentation is necessarily more

abbreviated Some statements whose pro ofs rely on deep algebraic or geomet

ric arguments are not explored in detail However the general strategy for

obtaining the desired algorithms from these results is explored fully

Constructions involving multivariate resultants haveplayed a large role

in the development of ecientsequential algorithms during the past decade

but only within the last few years have sp ecial cases of these results con

tributed signicantly to the improvement of parallel algorithm design The

uresultant of a set of n p olynomialsin n variables is p erhaps the most imp or

tant to ol here and we present a simple parallel algorithm for its computation

We also discuss the computation of socalled generalized characteristic poly

nomials and demonstrate how they aid in adapting resultants for the homoge

neous case to the inhomogeneous case These techniques have b een develop ed

recently in the design of parallel algorithms for deciding questions in the theo

ries of real closed and algebraically closed elds and for eliminating quantiers

in these theories Although an exp osition of such applications is b eyond the

scop e of this chapter wehave tried to gather together the fundamental results

of the mo dern and classical theories and present them in sucient detail to

enable the interested reader to pursue the more recentwork in this area

Many of the problems considered in this chapter are ultimately reduced

to computations in linear algebra such as the computation of determinants

or c haracteristic p olynomials A varietyofecient parallel algorithms are

known for these problems and are discussed in detail elsewhere in this volume

and in We also do not analyze the pro cessor eciency of these

algorithms This information can b e found in or in the references

Mathematical Preliminaries

This section presents much of the necessary mathematical background

from commutative algebra and ring theory necessary for the following sections

We assume some familiarity with linear algebra and parallel algorithms in

linear algebra as presented in The material on graded rings and

pro jective spaces is used later in x to develop the theory of multivariate

resultants and is not necessary for the understanding of univariate resultants

Omitted pro ofs in this section can b e found in any standard text on algebra

or algebraic geometrysuchas

Intro duction 

Unless otherwise noted k will denote an algebraically closed eld of

arbitrary characteristic In general lower case Roman letters will denote

variables and lower case Greek letters will denote elements of the eld k We

write x for a sequence of elements or variables x x Ifx is a sequence of n

n

E

variables and E e e isamultiindex we write x for the

n

e e

 e

n

x x denotes the ring of p olynomials in the IfR is a ring then R x x

n

x with co ecients in R variables

Polynomial Rings and Ideals

Let R k x x An ideal of R is a subset I of R closed under

n

addition and under multiplicationby elements of RAbasis for an ideal I is

a set of p olynomials B which generates I in the sense that every f I can

b e written

f g f g f

m m

for some f f B and g g R When an ideal has a nite

m m

basis wesaythatitisnitely generated and write f f for the ideal

m

generated by the p olynomials f f

m

THEOREM Hilbert Basis Theorem

Every ideal I R is nitely generated In other words every I is of

the form f f for some polynomials f f R

m m

Q P

n n

e

i

Dene the total degree of a monomial R to b e x e

i

i i i

The degree of a p olynomial f R is the maximum degree of anytermof f

We write R for the subset of all p olynomialsin R of degree at most e

e

e n

Each R is in fact a over k of dimension Wetakeas

e

n

a basis for R the set of all monic ie with leading co ecient

e

of degree at most e

e e

e



n

j e e e eg fx x x

n

n

Geometric Background

In this section we assume that k is an algebraically closed eld For

example k maybe C the eld of complex numbers

 Chapter Parallel Resultant Computation

n

We denote by A the space of ntuples elements of k called the n

x

dimensional ane space over k with coordinate functions x x x We

n

n

write A and omit the subscript when the co ordinates are understo o d For

f f R dene

m

n

V f f f k j f f g

m m

n

the set of common zeros of these p olynomialsin A A set of this form is

called algebraic A principle link b etween the geometry of algebraic sets in

n

A and the ideal structure of the p olynomial ring R is given by Hilb erts

Nullstellensatz or theorem of the zeros To state the Nullstellensatz in its

socalled weak form weletK b e an arbitrary subeld of the algebraically

closed eld k and consider the p olynomial ring R K x x R Let

n

us b egin with the case n where a pro of of the weak form of Hilb erts

theorem is more familiar Weknowthatany pair of univariate p olynomials

f and f have a common solution in k exactly when they haveanontrivial

greatest common divisor gcd ie a gcd which is a non p olynomial

Since K x is a Euclidean ring the Euclidean Algorithm for computing gcds

works here and implies that there exist additional p olynomials g g R

suchthat

g f g f gcd f f

So a necessary and sucient condition for the existence of a common zero

of f and f is that there are no p olynomials g and g R such that

g f g f The Euclidean algorithm itself provides a sequential decision

pro cedure in this case

The Nullstellensatz provides an analogue of this result for the case of

multivariate p olynomials in which the p olynomial ring R is not Euclidean and

the existence of a nontrivial gcd is not a necessary condition for the existence

of common zeros

THEOREM Hilberts Nul lstel lensatz weak form

Let n and let R K x x For f f Rthere exist

n m

elements k algebraic over K such that

n

f f

n m n

if and only if f f R

m

Intro duction 

From the denition of ideals we knowthat I R exactly when I

so as an immediate corollary of the Nullstellensatz we obtain the following

criterion for deciding when a set of p olynomialsin R has no common zero

THEOREM

Let f f R Then

m

V f f f f

m m

Nowiff f are p olynomialsin Rthenweknowthattheyhave

m

no common algebraic zeros exactly when f f or in other words

m

when there exist additional p olynomials g g R suchthat

m

g f g f g f

m m

The strong form of the Nullstellensatz tells us more ab out the relation

between the ideal I and its zero set V I

THEOREM Hilberts Nul lstel lensatz strong form

Let I R be an ideal and let f RThenf vanishes on every point

m

in V I if and only if f I for some m

This theorem will b e useful in dening a homogeneous analogue of the Null

stellensatz in the next section

Homogeneous Polynomials

The following facts are used only in the developmentofmultivariate

resultants in x b elow Our presentation parallels that of the previous

sections dening graded rings of homogeneous p olynomials and a homoge

neous version of the Nullstellensatz

Graded Algebras

A is a ring S together with a collection fS j e g of sub

e

groups of the additive group of S such that

L

S S and

e

e

L

The direct sum signies that every elementof S can b e written uniquely as a sum

P

f where each f S and all but a nite number of these f s are zero of the form

i i i i

i

 Chapter Parallel Resultant Computation

S S S for all d e

d e de

An element f S is called homogeneous if f S for some e For I S

e

an ideal dene I I S The ideal I is called a homogeneous ideal if it is

e e

generated by its homogeneous elements

M

I I

d

d

or equivalentlyif I has a basis of homogeneous elements x

Atypical example of a graded ringand the one on whichwe fo cus in

x is the p olynomial ring S Rx x which can b e graded in the

n

following way A p olynomial f Rx x issaidtobe homogeneous of

n

degree e if every term of f has total degree e and homogeneous if it is homoge

neous of some degree We dene S to b e the collection of all p olynomialsin

e

S which are homogeneous of degree e Since any p olynomial f S of degree

e can b e written uniquely as a sum

f f f f f

e

where each f is homogeneous of degree i it follows that S is a graded ring

i

When R is a eld it is a simple exercise to show that each subgroup S is

e

in fact a vector space over R In this case wetaketheset M of all monic

e

monomials of degree e

e e

e



n

M fx x x j e e e eg

e n

n

e n

as a basis for the Rvector space S We write n jM j

e e e

n

Pro jective Spaces

n

is the set of equivalence classes The ndimensional pro jectivespaceP

x

n

of p oints in A fg where

x

n n

n Note that if there exists a nonzero k such that i

i i

n n

each p ointin P is just a line through but excluding the origin in A To

Intro duction

more easily distinguish b etween p oints in ane and pro jective spaces we write

forthe equivalence class of ane p oints f j

n n

k and g

Let S k x x b e graded as in x Then f S is homoge

n

neous of degree e exactly when

e

f f

n n

for all and k This implies that the zero sets of homogeneous

n

p olynomials resp ect equivalence classes and it is meaningful to sp eak of the

p oints in pro jective space that are the zeros of a homogeneous p olynomial

Note that the ane p oint is a zero of every homogeneous p olynomial

and consequen tly has no counterpart in pro jectivespace

n n

The ane space A is emb edded in the pro jectivespace P under the

standardembedding

n n

n

The only p oints of P not in the image of this map are the p oints of the

socalled hyperplane at innity

f j not all i ng

n i

For any homogeneous ideal I S we will dene

n

V I f P j f forallf I g

n n

the zero set of I in P As in the ane case the geometry of P is related

to the ideal structure of the graded ring S by the Nullstellensatz nowina

homogeneous form

THEOREM Homogeneous Nul lstel lensatz

Let I S be a homogeneous ideal and let f S for some eThen

e

m

f vanishes on every point in V I if and only if f I for some m

As in the ane case we can use this theorem to dene necessary and sucient

conditions for the emptiness of V I for any homogeneous ideal I of S The

previous criterion that I is still sucient but no longer necessaryThe

Chapter Parallel Resultant Computation

dierence arises b ecause there is no pro jectivecounterpart of the ane p oint

THEOREM

Let f f behomogeneous polynomials in S and let I f f

m m

Then V I if and only if I S for some d

d d

In particular since S is generated by the M as a vector

d d

space over k itsucestoshow that all monomials of degree d are in I ie

that M I

d d

Univariate Resultants

In this section wedevelop the classical univariate or Sylvester resultant

an algebraic condition on the co ecients of a pair of univariate p olynomials

that determines whether they have a common ro ot

The Sylvester and the Resultan tof

TwoUnivariate Polynomials

Let k b e an algebraically closed eld and x an indeterminate Consider

twounivariate p olynomials f g k x of degree deg f and deg g resp ectively

deg f

X

i

x f x

i

i

deg g

X

i

g x x

i

i

Arrange the co ecients of f and g in staggered columns to form a square

matrix as in the following gure with deg g columns of co ecients of f and

deg f columns of co ecients of g The gure illustrates the case deg f

Univariate Resultants

and deg g

z z

deg g deg f

DEFINITION

The matrix is cal ledthe Sylvester or resultant matrix of f and g and

its determinant det is cal ledtheresultant of f and g

THEOREM

The univariate polynomials f and g have a common root in k if and only

if is singular ie if and only if the resultant of f and g vanishes

PROOF

Equivalentlywemust show that is singular i f and g haveanontrivial

gcd or in other words i the degree of the lcm of f and g is strictly

less than deg fg degf degg

Let k x denote the space of p olynomialsin k x of degree at most

d

d This is a vector space of dimension d over k with standard basis

d

xx The spaces k x k x and k x are

deg g deg f deg f deg g

b oth vector spaces of dimension deg f degg and under the standard

basis the matrix denotes the linear map

k x k x k x

deg g deg f deg f deg g

s t sf tg

 Chapter Parallel Resultant Computation

If is singular then the kernel of is nontrivial Thus there exist

nonzero s t with deg s deg g and deg tdeg f suchthatsf tg

Then divides sf tg so its degree is strictly less than deg f deg g

Conversely if deg deg f deg g then ker thus is

f g

singular

The resultantoftwo univariate p olynomials can b e computed in NC

using Csankys algorithm incharacteristic or Berkowitz or Chistovs

algorithm in arbitrary characteristic see

DEFINITION

Consider the coecients of f and g as indeterminates a a a

deg f

b b b Then the determinant of isapolynomial in k a b of

deg g

degree deg f deg g Thispolynomial is cal led the resultant p olynomial

For any sp ecialization of the indeterminates a b with and

deg f

giving p olynomials f g k x the value of the resultant p olynomial

deg g

on is the resultantoff and g

Subresultants Polynomial Remainder

Sequences and the Extended Euclidean

Scheme

An imp ortant application of resultants is in the calculation of the poly

nomial remainder sequences PRS that accrue from the execution of the

Euclidean algorithm Co ecients of elements of the PRS can b e expressed as

signed quotients of pro ducts of minors of the This holds

as well for the elements of the extended Euclidean scheme EES of whichthe

PRS forms a part Thus all co ecients of elements of the PRS and EES can

b e computed in NC In this section we describ e this algorithm and

prove its correctness

The basic fact on which the Euclidean algorithm rests is that for any

pair of p olynomials f and g there exist a unique quotient q k x and

remainder r k xsuch that deg r deg g and f qg r The Euclidean

f f algorithm calculates the sequence f f f g f f

n n

Univariate Resultants 

where for m n f is the remainder obtained by dividing f by

m m

f The p olynomial f is the last nonzero p olynomial in the sequence and

m n

is the gcd of f and g This sequence is known as the polynomial remainder

sequence PRS of f and g

DEFINITION

For m n let q k x be the quotient obtained in the division

m

of f by f thus

m m

f f q f

m m m m

Consider the polynomials s s s s and t t t t de

n n n n

nedby

s t

s t

s s q s t t q t

m m m m m m m m

for m n The col lectionofallthesepolynomials f s t

m m m

and q is known as the extended Euclidean scheme EES of f and g

m

The signicance of s and t is given in the following lemma

m m

LEMMA

deg f For m n Assume that deg g

i deg s degg deg f

m m

ii deg t deg f deg f

m m

iii s f t g f

m m m

Moreover s and t are the unique polynomials s and t such that

m m

a deg sdeg g deg f

m

b deg t deg f deg f

m

c sf tg and f have the same degree and leading coecient

m

PROOF

 Chapter Parallel Resultant Computation

Statements iiii are easily proved by induction on m These prop er

ties also imply that s and t satisfy ac Thus it remains to show

m m

uniqueness

We note that

deg g deg f deg f

m m m

for all m

deg s deg q s

m m m

deg s

m

deg s for all m

m

deg t deg q t

m m m

deg t

m

deg t for all m

m

Let s and t be any p olynomials satisfying ac Under assumption c

the statemen ts a and b are equivalent so we will only need to use

one Wehave

deg s sf tg deg s deg f by c

m m m

deg g byi

deg ss f t g deg s degf by iii

m m m

deg g bya

Subtracting weget

deg s sf tg ss f t g deg s t st g

m m m m m

deg g

e argument which is p ossible only if st s t But an easy inductiv

m m

shows that s and t are relatively primein fact

m m

m

s t t s

m m m m

Univariate Resultants

therefore there exists a p olynomial u such that s us and t ut

m m

Then

sf tg us f ut g

m m

uf

m

But by c it must b e that u therefore s s and t t

m m

At this p ointwe are ready to dene subresultants For d deg g

let b e the deg f degg d deg f deg g d submatrix of

d

obtained by deleting the last d columns of co ecients of f the last d columns

of co ecients of g and the last d rows The following gure illustrates

d

for deg f degg and d

d

z z

deg g d deg f d

Under the standard basis represents the linear map

d

k x k x k x

d deg g d deg f d deg f deg g d

d

s t the quotient obtained in the division of sf tg by x

DEFINITION

th

The matrix is cal ledthed subresultant matrix of f and g and its

d

th

determinant det is cal ledthed subresultant of f and g

d

THEOREM

d degf for some f in The matrix is nonsingular if and only if

m m d

the PRS In this case the vector of coecients of s and t forms the

m m

unique solution x of the nonsingular system

T

x a

d m

Chapter Parallel Resultant Computation

where a is the leading coecient of f

m m

PROOF

Note that d deg g Ifd deg f for any r let m b e the largest number

r

suchthatdegf dThen m n and by Lemma

m

deg s deg g deg f deg g d

m m

deg t deg f deg f deg f d

m m

deg s f t g deg f d

m m m

so s t ker therefore is singular

m m d d

Now supp ose that d degf Then s t a the leading co ef

m d m m m

cientoff therefore the vector x of co ecients of s and t satises

m m m

Moreover by Lemma s and t are unique therefore

m m d

is nonsingular

This theorem gives rise to an NC algorithm for calculating all elements

of the EES

ALGORITHM

Extended Euclidean Scheme

Input Given p olynomials f and g

Output The extended Euclidean scheme for f and g

th

For each d deg g compute the d subresultant det Thed for which

d

is nonsingular are exactly the degrees of the f in the PRS

d m

For each d degf m solve the nonsingular system

m

T

x

d

s t

m m

This gives the coecients of s and t We do not yet

m m

a a

m m

know a

m

f

m

Compute f s f t g This is the monic associate of f

m

m m m

a

m

Divide f by f using Algorithm below Alternatively solve

m m

th

using the d subresultant matrix of f and f

m m

Univariate Resultants 

This gives a constant b and polynomial p such that

m m

b f f p f

m m

m m m

Compute

a b b b b m even

m

a

m

a b b b b m odd

m

a

m

p q

m m

a

m

f a f

m m

m

s a s

m m

m

t a t

m m

m

The computations in Step are justied by the following argument In

Step we computed b and p such that

m m

b f f

m m m

f p

m m

a a a

m m m

thus by the uniqueness of quotient and remainder

b a a

m m m

f f p f

m m m m

a a

m m

f q f

m m m

f

m

whence follow and the recurrence

a b a

m m m

with solution

The solution vector to computed in Step is the last column of

whichby Cramers rule is the last column of the adjointof divided

d

d

by det This indicates that all the co ecients of s and t are signed

d

m m

quotients of minors of the Sylvester matrix

Algorithm can b e implemented in NC using standard to ols for linear

algebra see

 Chapter Parallel Resultant Computation

Polynomial Division with Remainder

Let f g b e p olynomials deg g deg f and let q and r b e the quotient

and remainder resp ectively obtained in the division of f by g Algorithm

suggests an NC algorithm for p olynomial division with remainder compute

the subresultants to nd d degr then solve with

d

However this algorithm has two serious liabilities

It requires the computation of all the subresultants

It requires divisions in k

The latter b ecomes a ma jor problem when the co ecients of f and g are inde

terminates Algorithm expresses the co ecients of q and r as quotients

of p olynomials in the co ecients of f and g This is so even when the divisor

g is monic in which case the co ecients of q and r are p olynomial functions of

the co ecients of f and g rather than rational functions In the computation

of the multivariate resultant to b e presented in x it will b e essential

to have a p olynomial division algorithm for monic g that do es not use an y

divisions in k but only the ring op erations and

Here wegive a resultantstyle algorithm used byCanny that alleviates

these problems The algorithm is based on the following theorem

THEOREM

Let m degf deg g the dimension of Ifg is monic

deg g

mi

then the coecient of x in q is

i

i

i m det

deg g

i

th

where det is the i principal minor determinant of the upper

deg g

left i i submatrix of

deg g

PROOF

Let

deg q deg f

X X

i i

x q x f

i i

i i

deg g deg r

X X

i i

g x r x

i i

i i

Univariate Resultants

Note that deg q m The equation r f qg is expressed in the

deg f m linear system

here illustrated for the case deg f degg and deg r The

rst m rows of this matrix comprise

deg g

Now consider the m m system obtained by taking the rst m rows

of and last row

and let A b e the m m matrix in Certainly A is nonsingular

m

since its determinantis Theinverse of A is given by Cramers

th

rule the i j entry of A is

det A

ji

ij m ij

det A

ji

det A

where A is the m m submatrix obtained from A by

ji

th th

dropping the j rowandi column In particular the last column of

T

whichby is the vector contains A

m

i

det A i m mi

 Chapter Parallel Resultant Computation

But note that for this particular matrix

i

det A det A

mi

i

det i m

deg g

Thus for i m

i

i

det

mi

deg g

i

i

det

deg g

Using Theorem we can give the following simple divisionfree algo

rithm for p olynomial division with remainder when the divisor is monic

ALGORITHM

Polynomial Division with Remainder

Input Polynomials f and g degf deg g g monic

Output Polynomials q and r suchthatf qg r

Compute the principal minors of

deg g

Set

i

i

i m det

mi

deg g

j

where m deg f deg g Then is the coecient of x in q

j

Set r f qg

is not monic then divisions are inevitable However we can apply If g

Algorithm to the monic asso ciate of g then adjust q afterward by dividing

by the leading co ecientofg

All op erations can b e implemented in NC The principal minors of

can b e computed without division using Berkowitz or Chistovs

deg g algorithm

Univariate Resultants 

A Resultant System for Several Univariate

Polynomials

The constructions of x and x can b e mo died to yield NC

algorithms for testing whether a set of univariate p olynomials has a common

ro ot and for computing their gcd We reduce these problems to the case

of two univariate p olynomialsover a larger eld In x b elow wewill

show that in the presence of a source of randomness these algorithms have

implementations that are no less ecient than the algorithms of x and

x for two p olynomials

Let f f k x Let y b e a new indeterminate and consider the

n

bivariate p olynomial

n

f x y f xf xy f xy f xy

n

We regard f as a p olynomial in the indeterminate x with co ecients in the

transcendental extension k y ofk Thus it makes sense to consider the gcd

of f and f over k y x

n

LEMMA

The gcd of f and f is the same as the gcd of f f In other words

n n

f f and f f generate the same principal ideal in k y x

n n

PROOF

Let g b e the monic gcd of f f in k x and let h b e the monic gcd

n

of f and f in k y x Certainly g divides hsince g divides f and f

n n

To show that h divides g it suces to show that h divides f f

n

Since h divides f and h is monic h k x For i nletq and r

n i i

b e the quotient and remainder resp ectively obtained in the division of

f by h in k x Then f qh r where

i

n

X

i

q y q

i

i

n

X

i

r r y

i

i

and the degree of r as a p olynomial in k y x is less than deg h Since h

divides f in k y x r mustbeidentically zero as a p olynomial in k y x

Chapter Parallel Resultant Computation

Since y is transcendental r is also identically zero as a p olynomial in

k x y Thus r and h divides f i n

i i

Now form the Sylvester matrix of f and f Theentries of are

n

p olynomials in k y of degree at most n and the resultant detisa

p olynomial in k y of degree at most n deg f

n

THEOREM

The polynomials f f have a common root in k if and only if det

n

vanishes identical ly

PROOF

The p olynomials f f have a common ro ot in k i they havenon

n

trivial gcd By Lemma this o ccurs i f and f haveanontrivial

n

gcd in k y x ie if f and f have a common ro ot in the algebraic

n

closure of k y By Theorem this o ccurs i the resultantoff and

f vanishes identically

n

This gives rise to the following NC algorithm

ALGORITHM

Resultant of Several Polynomials

Input Polynomials f f k x

n

Output The resultantoff f

n

Let y be a new indeterminate Form the polynomial

n

X

i

f x y f y

i

i

For eciency it makes sense to take f of minimum degree among

n

f f

n

Form the Sylvester matrix of f and f The entries arepolynomials

n

in k y of degree at most n

Univariate Resultants

Calculate the resultant det of f and f using Berkowitz or Chistovs

n

algorithm This is a polynomial in k y of degree at most n

deg f

n

Check whether det vanishes identical ly By Theorem this occurs

i f f haveacommon root

n

As in x if the co ecients of f f are indeterminates then

n

Algorithm can b e carried out symb olically The entries of the Sylvester

matrix of f and f are then p olynomials in y and the indeterminate co ecients

n

a of the f The resultant is a p olynomial r a y of degree at most n

i

n

deg f in y and deg f max deg f in the a Considering r a y asa

n n i

i

a Theorem implies that all these p olynomial in y with co ecients in k

co ecients vanish under a sp ecialization of a i the p olynomials f f

n

have a common ro ot in k The co ecients resulting from the sp ecialization

of r aythus form a resultant system for f f

n

It is p ossible to work in the p olynomial ring k x y a explicitly and com

pute the symb olic resultant r a y inNC using Berkowitz or Chistovs

algorithm These algorithms pro duce a p olylogdepth p olynomialsize circuit

a y over constants and inputs a and y For any sp ecialization C for r

of a since r y is of degree at most n deg f we can test in NC

n

whether r y vanishes identically byevaluating it at n deg f

n

sample elements of k using the circuit C

The GCD of Several Univariate Polynomials

The construction of x can b e extended to give a deterministic NC

algorithm for computing the gcd of several p olynomials Wewillshowin

x b elowhow to improve the eciency in the presence of a source of

randomness

ALGORITHM

GCD of Several Polynomials

Input Polynomials f f k x

n

Output g k x the gcd of f f

n

Chapter Parallel Resultant Computation

Let y be a new indeterminate and form the polynomial

n

X

i

f x y f y

i

i

For eciency take f of minimum degree among f f

n n

Form the subresultant matrices of f and f The entries of are

d n d

polynomials in y of degreeatmostn

Compute the subresultants over k y x using Berkowitz or Chistovs

th

algorithm The d subresultant det isapolynomial in y of degree

d

at most n deg f d

n

Let d be the smal lest number such that det does not vanish identical ly

d

This is the degree of the gcd of f and f

n

Compute the monic gcd of f and f as in Algorithm using polyno

n

mial arithmetic on the coecients By Lemma this is also the gcd

of f f

n

Reduce the coecients As computedinSteptheyarerational func

tions of y ie quotients of polynomials in y but they reduce to elements

of k because the monic gcd is in k x

Improving Eciency with Randomness

If wehave access to a source of randomness then we can obtain signi

cantly more ecient algorithms than those of x and x by using a

randomly chosen elementofk in place of the indeterminate y This is in fact

the metho d of c hoice in most implementations This approach is based on the

observation that a nonzero p olynomial is not likely to vanish when evaluated

on a random input chosen from a suciently large sample set

This idea is made concrete in the following lemma proven indep endently

byZippel and Schwartz

LEMMA

Let px x be a nonzeropolynomial of total degree d with coe

n

cients in k and let S be a nite subset of k Ifp is evaluatedonarandom

Univariate Resultants

n

element s s S then the probability that ps s is

n n

d

at most

jS j

This gives rise to the following probabilistic NC algorithm

ALGORITHM

Resultant of Several Polynomials Probabilistic Version

Input Polynomials f f k x

n

Output The resultantoff f

n

Select a random element uniformly from a large nite set S k

Form the polynomial

n

f xf x f x f x

n

This is f x where f is the polynomial

x and f xIff f have a com Calculate the resultant r of f

n n

mon root then r with probability If f f do not have a

n

common root then by Lemma r with probability at most

n deg f

n

jS j

Reducetheprobability of error by repeated trials

This algorithm do es not provide a waytocheck the accuracy of the

output This liability is corrected in the gcd algorithm b elow

As shown in Lemma if g is the gcd of f f in k x then

n

g is also the gcd of f and f in k y x where f is the p olynomial

n

With high probability g will also b e the gcd of f and f x for a random

n

chosen uniformly from a suciently large subset of k This is b ecause g

always divides the gcd of f and f x and by Theorem g is not itself

n

th

x vanishes the gcd of f and f x i the d subresultantoff and f

n n

th

where d deg g Thisisthe d subresultantoff and f x y evaluated at

n

th

and again by Theorem the d subresultantoff and f x y do es not n

 Chapter Parallel Resultant Computation

vanish identicallythus Lemma applies This gives the following RNC

algorithm

ALGORITHM

GCD of Several Polynomials Probabilistic Version

Input Polynomials f f k x with gcd of degree d

n

Output The p olynomial g gcdf f

n

Select a random element uniformly from a large set S k

Form the polynomial

n

f xf x f x f x

n

This is f x where f is the polynomial

Calculate the gcd h of f x and f as in Algorithm Then g

n

th

divides hand h does not divide g i the d subresultant of f x and

f vanishes By Lemma this happens with probability at most

n

n deg f d

n

p

jS j

Check whether h divides g by checking whether h divides f i n

i

using Algorithm If so h is the desired gcd Ifnotgoback to Step

andrepeat with a new random

Using the value p in as a b ound on the probability of failure in

each trial and assuming the trials are indep endent the exp ected number of

trials b efore successfully obtaining the gcd of f f is at most

n

p

See von zur Gathen for another approach attributed therein to S

Co ok which yields a deterministic NC algorithm

Multivariate Resultants

The resultant of univariate p olynomials is a classical to ol that has played

a considerable role in mo dern algorithms in symb olic algebra and computa

tional geometryItprovides an eectively computable algebraic criterion for

Multivariate Resultants 

deciding when twoormoreunivariate p olynomialshave a common ro ot Quite

early in this century eective means were develop ed for generalizing the re

sultant to the case of multivariate p olynomials Here however there must

b e a somewhat dierent approach Chevalley had noted that there can b e no

strictly algebraic criterion for the existence of common solutions to a system

of multivariate p olynomials in the sense that the pro jection of an algebraic

set is not necessarily algebraic

Because there is no purely algebraic criterion classical attempts at algo

rithms for the multivariate case diverge One direction saw the development

of resolvents through iterated application of the classical univariate resultant

to eliminate variables one at a time In the work of Hermann Kronecker

Macaulay and others an algebraic criterion was found for the sp ecial case

where the given p olynomials are all homogeneous Here one is dealing with

the existence of common zeros in pro jective space The solution whichis

b oth elegant and reasonably ecient parallels and generalizes the basic the

ory develop ed in x for the univariate case

In this section we review some of the basic prop erties of multivariate

resultants and their computation We do not present detailed pro ofs but

instead state the prop erties of various algebraic ob jects and give construc

tions of such ob jects whichhave b een found useful in the design of algebraic

algorithms

Below S denotes the ring k x x of p olynomials in n variables

n

with co ecients in an algebraically closed eld k

Resultant Systems

Classical elimination theory considered b oth necessary and sucient

conditions for homogeneous p olynomials f f to have no common pro

m

jective zeros Recall that the Homogeneous Nullstellensatz asserts that this

E

o ccurs exactly when for some devery monomial x of degree d can b e written

E

x g f g f g f

m m

for some p olynomials g g S An eective criterion for deciding

m

whether the f s have a common pro jective zero can b e derived by nding

i

a b ound on this degree d and on the degrees of the g s whichmust exist

j when there are no solutions

 Chapter Parallel Resultant Computation

An initial solution to this problem is provided by the next theorem It is

Lazards mo dern generalization of a classical result of Kronecker proved

using homological metho ds

THEOREM Eective Homogeneous Nul lstel lensatz

Let m n and f f S k x x behomogeneous

m n

P

n

polynomials generating the ideal I Let d deg f Then

i

i

V f f if and only if I contains every monomial of degree d

m

When m n it is known that V f f isnever empty a fact which

m

follows from Krulls Hauptidealsatz and the Pro jectiv e Dimension Theorem

x When m n this degree b ound is tight For example the

d d

 d

n

p olynomials x have no common pro jective zeros although the x x

n

d d

d

 

n

ideal generated by them do es not contain the monomial x x x

n

P

n

d d It do es however contain every monomial of of degree

i

i

degree d

We derive a parallel algorithm to verify this condition by reduction to

a problem of linear algebra Recall that the additive subgroup S of S is a

d

vector space over k with basis M By Theorem to determine whether

d

V f f it is b oth necessary and sucien ttoshowthatevery

m

E

monomial x M is in the ideal generated bythe f s Using the following

d i

Lemma we reduce this verication to a problem of linear algebra

LEMMA

Let f f behomogeneous polynomials and I f f Let

m m

d deg f i m

i i

If h I then thereare homogeneous polynomials g g with g

d m i

S such that h g f g f g f

dd m m

i

PROOF

The lemma is proved by noting that if h I then there are p olynomials

d

P

m

g xsuchthath g k g f Observe that all terms of

i

m i

i

which are not of of degree d d give rise to terms that must b e g

i

i

cancelled in the summation So if wetake g to b e the part of g that is

i

i

P

m

g f as well homogeneous of degree d d thenh

i i i

i

Multivariate Resultants 

It is easy to see that as a consequence of Lemma the map which

takes every vector of m homogeneous p olynomials g g g S to

m i dd

i

P

m

the sum g f

i i

i

m

Y

S S

dd d

i

i

m

X

g g g f

m i i

i

is a k linear map of vector spaces From Lemma it also follows that the

image of is exactly I We can now dene the matrix of the map with

d

resp ect to the bases M and M i m analogous to the matrix of

d d

i

e index the rows of this matrix by the elements of M and the x W

d

B B

columns by pairs i x where i m and x M The column

dd

i

Q P

m m

indices comprise a basis for S of size n Theentry of

dd dd

i i

i i

A B A

in row x and column i x is just the co ecient of the term x in the

B

x f x When n and m gives the Sylvester matrix p olynomial

i

of the two univariate p olynomials f x andf x dened in

x

The Eective Homogeneous Nullstellensatz Theorem now implies

that V f f if and only if is surjective This happ ens exactly

m

when the matrix has full rank jM j n Soonewaytoverify that the

d d

given system of p olynomials has no solution is to nd a nonzero n n minor

d d

of which exists if and only if has rank n

d

The multivariate resultant allows us to eliminate variables from some

collections of multivariate p olynomials Supp ose that f x y f x y

m

are p olynomials in two sets of variables x x x and y y y

n n

x and in addition that they are homogeneous as p olynomials in the variables

Construct the matrix y with resp ect to xsothattheentries of y are

p olynomials in y Then the matrix y has the following prop erty for any

n

p oint A the system f x f x has a solution in the variables

m

x if and only if has rank strictly less than n x Thus

d

n n

A j P f f g f

m

 Chapter Parallel Resultant Computation

n

d

f A j rank n g

n

f A are zero g j all n n minors of

d d

As in x instead of concentrating on a single set of p olynomial equations

we can go one step further and regard the co ecients as parameters This

allows us to compute a general algebraic criterion expressed in terms of the

indeterminate co ecients One can in principle compute a resultant system

consisting of a set of p olynomials in the indeterminate co ecients then sim

ply evaluate them on the co ecients of anygiven system and immediately

determine whether or not a solution exists Unfortunatelythenumber of

p olynomials which arise and their degree make the computation unrealisti

cally complex in all but the most trivial cases Nevertheless we continue to

discuss the computation of resultant systems in these terms for several rea

sons First there are in fact some essential results from classical elimination

theory that require this form In addition this form allows us emphasize that

the algorithms whichwe present can b e expressed solely in terms of the basic

op erations of the ring of co ecients and hence commute with substitution

Finally it p ermits us to express the complexity of the quantities that arise in

this computation in a purely algebraic manner in terms of the complexityof

each co ecient

Let f f b e a set of homogeneous p olynomials in the variables

m

x x with indeterminate co ecients among the variables c In other

n

P

A

c x x words we are considering p olynomials f of the form f c

i i iA

A

A

where A ranges over multiindices of some xed degree d each x is a mono

i

mial of degree d and each co ecient c is a distinct indeterminate

i iA

DEFINITION

A resultant system for the polynomials f f is a col lection of

m

polynomials g g k c with the fol lowing property for every spe

r

c of the coecients to elements of k the polynomi cialization

n

als f xf x have a common solution in P if and only if

m

g i r In other words

i

n

j P f f g V g g f

m r

If r this polynomial is cal led a resultant for the system f f

m

Multivariate Resultants

Additional details can b e found in the texts of Macaulay Ch and van

der Waerden Ch

Under current technology the direct calculation of such resultant sys

tems by computing all minors of the corresp onding matrix c is infeasible

b ecause of the large numb er of p olynomials and their high degree Toim

prove this situation somewhat we will use the parallel algebraic matrix rank

algorithm of Mulmuley We summarize the relevant asp ects of the con

struction in the statement of the next lemma A more complete treatment

can b e found in or

LEMMA Mulmuleys Rank Algorithm

Let A beanm n matrix over an arbitrary eld k m n and let z w

be indeterminates It is possible to compute a polynomial p k w z of

degree at most m n in w and n in z such that the rank of A is

nj

j

where z is the highest power of z that divides pInparticular

A is of ful l rank if and only if j ie if and only if pw does

not vanish identical ly Moreover the computation uses only the ring

operations of the eld k and can be implemented by an arithmetic circuit

of size polynomial in m n and depth O log m n

Since Mulmuleys algorithm uses only the ring op erations of k it can also b e

p erformed on matrices containing indeterminate entries As a consequence

sp ecialization of these indeterminates to elements of k give the same result as

rst p erforming the substitution and then applying the algorithm

As a result we get the following theorem whichprovides an eective crite

rion for determining when there exists a solution to a system of homogeneous

p olynomial equations

THEOREM

Let f f k y x bepolynomials that are homogeneous in the

m

variables x of degree d d respectively with coecients in k y A

m

necessary and sucient algebraic condition for the existenceofacom

n

mon zero P expressed in terms of the parameters y can becomputed

in paral lel polynomial time with respect to the elementary operations of

 Chapter Parallel Resultant Computation

the ring k y In other wordswecan compute a set of polynomials

g g k y such that

r

n

g g P f f

r m

PROOF

Supp ose f f are as describ ed in the statement of the theorem

m

Let b e the matrix of the linear map of Then is an n

d

P

m

n y To compute a resultant system we matrix over the ring k

d

i

i

will apply the parallel matrix rank algorithm of Mulmuley to From

P

e

j

q y w y w that this algorithm we obtain a p olynomial q

j

j

for every substitution of eld elements for the variables y is identically

zero as a p olynomial in w if and only if do es not have full rank n

d

y wisidentically zero just when all of its As a p olynomial in w q

co ecients are zero The collection of co ecients f q y j i e g

i

thus comprises a resultant system for the given set of p olynomials Since

the computation involves only the op erations of the co ecient ring the

computation also commutes with sp ecialization of co ecients

n

If each d d then the entire computation requires O d pro cessors and

i

time O n log d The numb er of p olynomials in the system and their de

m

d gree are at most exp onential in n and max

i

i

The Resultantofn Polynomials in n

Variables

When the numb er of homogeneous p olynomials equals the number of

variables there is also a single resultant p olynomial The presentation in this

section follows along classical lines For an excellent complete development

when k C which uses only elementary arguments see Renegar

LEMMA

For n homogeneous polynomials f f in the n variables

n

c there exists a single resul x x with indeterminate coecients

n

tant polynomial r c which can beeectively constructedfrom the matrix

Multivariate Resultants 

P

n

d and n jM j Let deg f d i n and dene d

i d d i i

i

Recall that the matrix has n rows each indexed by a monomialin M

d d

To construct the resultant p olynomialwe rst partition M into n sets

d

d

A A

i

Say that a monomial x M is reducedinx if x is not divisible by x

d i

i

i

For each i i n dene M to b e the set of monomialsin M that

d

d

d

i

are divisible by x and reduced in the variables x x Then the sets

i

i

n

M M comprise a partition of M

d

d d

Let A b e the square submatrix of obtained by selecting the columns

E E

i

x for i n and x M It can b e shown that of lab eled byi

d

as a p olynomial in the indeterminate co ecients cdetA is divisible bythe

resultant r of the f s Moreover the quotient of the determinantofA bythe

i

resultant r is itself the determinant of a submatrix B of A obtained by

E

eliminating those columns in A corresp onding to indices i x where

E

the monomial is reduced in n of the variables x x and

E

for each such column i x removed in Step eliminating the row

d

i

in this column containing the co ecientofx

i

These facts are proven by Macaulayin It follows that the desired resul

c by constructing tant can b e computed as a p olynomial in the indeterminates

the matrices A and B from and nding the quotient r detA det B When

constructed in this manner the submatrix B dep ends only on the co ecients

of the p olynomials f j f j and for each i r c is homogeneous

x n x

 

Q

of degree d in the co ecients of f In Ch Macaulay also shows

j i

j i

that r is irreducible as a p olynomial in k c See also Ch for further

details

Calculation of the resultant as presented ab ove requires a computation

in which all of the co ecients are indeterminates In most cases this com

P

n

n suchcoecients putation is prohibitively exp ensive since there are

d

i

i

Most often we do not need to know r as a p olynomial in the cs but are inter

ested only in the image of r under some substitution for these co ecients For

example let us assume that we wish to compute the resultant of the homoge

neous p olynomials f f k x x where all of the co ecients are

n n

constants in the eld k Note that all of the ab ove calculations that require

 Chapter Parallel Resultant Computation

only ring op erations commute with substitution for the indeterminates Ch

since substitution determines a ring homomorphism The determinant for

example is dened and computed using only ring op erations so we can either

compute det A as a p olynomial in k c and then sp ecialize the variables c to

the co ecients of the f s or sp ecialize the co ecients and then compute the

i

determinant In b oth cases the result will b e the same

The only other op eration required in the construction is the division

of these determinants Unfortunately this op eration do es not commute with

sp ecialization For example where all co ecients are elements of the eld k

as ab ove it is p ossible that det A detB under the given substitution

while the resultant itself is nonzero Toovercome this obstacle wemodify

the construction as follows Observe that the co ecients c of the terms

i

d

i

x in f lie only on the diagonals of the matrices A and B and that these

i

i

comprise all diagonal entries Instead of computing the determinants of A

and B directlywe compute the characteristic polynomials of these matrices

ie the determinants

a c t det tI A

bc t det tI B

where t is a new indeterminate These determinants are obtained from the

determinants of A and B by substituting t c for the co ecient c of the term

i i

d

i

in each f and negating all other co ecients It follows that b divides a c x

i i

i

and the quotient r c t is just the resultant of the new system

d

d



n

f f tx c x tx c x

n

n

tnorb tvanishes identically Moreover it is easy to see that neither a

n

for any k This implies that

b t r t a t

t is the resultant of the system under the sp ecied substitution so r

and r the resultant of the original system f f It is easy to see

n

that negation of the co ecients do es not change the resultant Hence we can

compute the resultantofn homogeneous p olynomialsin n variables

Multivariate Resultants 

with co ecients in a eld k by computing the of the quotient

of the characteristic p olynomials of A and B

As noted in x computation of the quotientoftwounivariate p oly

nomials suchasa tandb t can b e p erformed in parallel using only the

ring op erations of the co ecient eld when the divisor is monic We claim

further that the quotient of the characteristic p olynomials ac tandbc t as

c t of the p olynomials in polynomials in t is in fact the resultant r

For supp ose this were not the case and let r b e the quotientofa and b as p oly

nomials in tItmust b e that the actual resultant r divides r Butifr r

then r must have an additional factor p csuchthatr c t r c tpc So

ac t bc tr c tpc

and since

d

X

d i

c t t c t a a

i

i

it is clear that p

Hence r c tmust b e the the resultant of and the constant

term r c of this p olynomial is always the resultant of the original system

f f Since the constant term of this quotient can b e computed using

n

only ring op erations the entire computation describ ed ab ovenowcommutes

with sp ecialization of the indeterminate co ecients In other words wecan

rst substitute elements of any division ring for the co ecients of the f s and

i

then p erform the indicated computation The result is guaranteed to b e the

same as would b e obtained by constructing the actual resultant p olynomial

and then p erforming the same substitution

This discussion suggests the following parallel algorithm for computing

resultants Let f f b e p olynomialsinthevariables x x with

n n

co ecients in a ring R Construct the matrices A and B and compute their

characteristic p olynomials a b R t as discussed in For example R

y Write may b e the p olynomial ring k

n n

d d

a t a t a at t

n

d

e e

bt t b t b t b

e

 Chapter Parallel Resultant Computation

Q

n

d and compute the division ab using Algorithm where e n

i d

i

Since b is monic only the ring op erations are needed This gives the desired

resultant

See also x for another metho d of computing these classical re

sultants

THEOREM Resultant of n polynomials in n variables

The resultant of a set of n homogeneous polynomials f f in n

n

variables x x with coecients in a ring R K y can becomputed

n

in paral lel polynomial time relative to the elementary operations of the

ring R Moreover sincethecomputation uses only ring operations on

the coecients the calculation commutes with substitution

The computation of the characteristic p olynomials a and b require op

erations on matrices of size n Using Chistovs algorithm see this can

d

b e done in parallel time O log n in the elementary op erations of the co e

d

cient ring R The quotient of these two p olynomials requires computing the

Q

n

determinant of a matrix of size n e d and so can b e executed in

d i

i

Q P

n n

d O log d again measured in terms of parallel time O log

i i

i i

the elementary op erations of the ring of co ecients R We can conserv atively

b ound the size of elements in R that arise in this computation by noting that

Q

the co ecients of a are p olynomials of degree deg d in the co ecients

i

j i

of each f For example assume that f f are integral p olynomials with

i n

maximum degree d and let c b e a b ound on the numb er of bits necessary

to express any one co ecient Then the co ecients of a require fewer than

n n

n d c bits and those of the resultant no more than n d c bits On

the other hand if the co ecients of the f s are p olynomials in the ring k y

i

x and e in y then the co ecients of a are and have maximum degree d in

n

p olynomials of degree no more than n d e in y and those of the resultant

n

of degree at most n d e in y

The uResultant

The uresultant is a classical to ol for solving systems of homogeneous

equations whichhave only a nite numb er of pro jective solutions Its use has

Multivariate Resultants 

b een revived by computational applications as in the zeronding algorithm

of Lazard the approximation algorithm of Renegar and the work

of Canny in theoretical rob otics and of Renegar in real algebraic

geometry

Behind the construction of uresultants lies the following idea Sup

p ose that wehave n homogeneous p olynomial equations f f in the

n

n variables x x with only a nite numb er of pro jective solutions

n

s

n

P Then for almost every additional p olynomial f whichwe

might add to this system the enlarged system will have no common zeros

This is true even if the degree of f is constrained to b e

More concretely let u u b e new indeterminates and let f b e the

n

P

n

linear form u x We shownow that for most assignments of u x

i i

i

n

values P to u the system xf f has no common zeros Construct

n

the resultant r uofthese n p olynomials as a p olynomial in the variables

u By the characterization of the resultantgiven in x we know that r

n

is homogeneous in u and that for any p oint P r if and only if

f f and x have a common solution Equivalently r ifand

n

j

for some j This means that only if

s

Y

j

A

V r V u

j

Q

j

s

u The Homogeneous Nullstellensatz nowsays that eachofr and

j

divides some p ower of the other therefore r factors as a pro duct of linear

j

j

forms eachoftheform u for some zero of the f s Hence all of the

i

zeros of the f s can b e recovered by computing this resultant r and factoring it

i

over k u The co ecients of these factors are the co ordinates of the common

zeros

DEFINITION

Let f f k x x be homogeneous polynomials generating a

n n

zerodimensional ideal and let u u u be a set of new indeter

n

u x with minates Then the resultant of the f s and the polynomial

i

x is cal led the uresultant of the f s respect to the variables i

 Chapter Parallel Resultant Computation

The classical theorem on the uresultant asserts that when the set V

V f f is nite the p oints V and their multiplicities can b e recov

n

ered from a factorization of the p olynomial r u

LEMMA The uResultant

Let f f k x x behomogeneous polynomials and assume

n n

that V V f f is nite whereeach V has multiplicity

n

Thentheuresultant r u is a of degree

Q

n

deg f and

i

i

n

X Y

u r u

i i

i

V

Note that since the s are p oints in pro jective space this p olynomial is

unique only up to a nonzero constant factor

The argumentsketched ab ove can b e extended to showthat r u

if and only if V f f is nite From Lemma it follows that this

n

uandbu By the same lemma uresultant is a quotient of determinants a

we can construct these p olynomialssothat b is indep endent of the co ecients

of one of the given p olynomials In particular we can construct a and b from

the f s and the p olynomial u x so that b is indep endentofthevariables u

i

Thus when f f k x b is a constantink Whenever b a

n

u r u so that a is itself a uresultant p olynomial In this case the and a

uresultant can thus b e constructed using a single determinant computation

over k u

Generalized Characteristic Polynomials

The algorithms develop ed in x and x for computing resul

tants and uresultants to ok advantage of the fact that the quotientofthe

characterisitic p olynomials of the matrices A and B has two imp ortant fea

tures it never vanishes identically and it is the resultanturesultant of a

We write f g to signify that f and g are equal up to a nonzero constant factor ie there

is a k  such that f g

Multivariate Resultants 

p erturbation of the original system of equations by a new indeterminate This

observation was used to pro duce an ecient algebraic algorithm for computing

the resultantofn p olynomials in n variables directly from their co ecients

using only the op erations of the co ecient ring This quotientwas given the

name generalized characteristic polynomial byCanny b ecause it sp ecializes

in the case of linear equations to the characteristic p olynomial of the matrix

of the linear system

Recently such p olynomials have found wide use in extending resultants

and uresultants to the case of ane inhomogeneous sets Let us b egin

by considering how the algorithms of x might b e adapted to handle

inhomogeneous p olynomials Recall the standard emb edding of ndimensional

ane space into ndimensional pro jective space

n n

We can exploit this corresp ondence in the following manner Let f R be a

p ossibly inhomogeneous p olynomial in the n variables x x and let x

n

h

b e a new variable We dene the homo genizationq of f written f tobethe

p olynomial

deg f

h

f x x x x f x x x

n n

where deg f is the total degree of f Op erationally this means that wemul

tiply each term of f by a suciently large p ower of the new variable x to

bring it up to degree deg f

Although homogenization gives an op erational way of obtaining a related

homogeneous p olynomial from an inhomogeneous one we still have not shown

that there is a relation b etween the ro ots of these p olynomials whichcanbe

exploited in resultantbased algorithms On the one hand the reader can

n

easily verify that A is a zero of f exactly when

n

h h

yhave zeros whichdo is a zero of f However the p olynomial f ma

n

not corresp ond to zeros of the original p olynomial when weset x Such

solutions lie on the hyp erplane at innity and are called improper or innite

Let us consider the case of the uresultant Weknowthata uresultant

p olynomial can b e computed for n homogeneous p olynomials in S whenever

they have only nitely many pro jective zeros Now supp ose we are given n

p ossibly inhomogeneous p olynomials f f R and wish to compute their

n

Chapter Parallel Resultant Computation

h h

uresultant We homogenize them and compute the uresultantoff f

n

This uresultant should b e a constantmultiple of the p olynomial

n

Y Y X

A A

u u u

i i

i

where ranges over all common zeros of f f and ranges over all im

n

h h

prop er solutions of f f and from it we can still recover the common

n

zeros of the original system But this is only p ossible when the number of im

prop er solutions is nite For if there are innitely many improp er pro jective

solutions the uresultant p olynomial vanishes identically and no information

ab out the prop er solutions will b e available

What wewould like is a guaranteed metho d of obtaining all prop er

solutions to the original systemp erhaps together with a nite number of

the improp er solutionsin the manner that the uresultantprovides for the

strictly homogeneous case This is given by the following lemmaproved by

Canny for the case k C and by Ierardi Ch for elds of arbitrary

characteristic We state the lemma only in its simplest form

LEMMA

Let f f be inhomogeneous polynomials in n variables with only a

n

nite number of common zeros Then the uresultant of the system

deg f

h deg f h

n

f x tx x tx f

n n

with respect to the variables x x is a polynomial

n

d

X

i

ut ut r r

i

ic

u factors as in which the least nonzerocoecient r

c

n

Y X Y

A A

u x u

i i

i

Multivariate Resultants 

Q

n

where deg f ranges over al l points in V f f Here d

i n

i

and the points correspond to certain points in the algebraic set which

lie on the hyperplane at innity

Applications of uResultants

Many of the applications of resultants and uresultants arise from the

p ossibilityofrecovering the co ordinates of p oints dened as the zero set of a

number of multivariate p olynomial equations To a large extent the eciency

and elegance of these parallel algorithms stems from their ability to recover

information ab out these p oints symbolical ly without resorting to numerical

approximation

For example supp ose that a uresultant p olynomial r u has b een con

structed by one of the metho ds outlined in x and we wish to compute

th

the i co ordinate of the ane p oints represented by this form One way

of obtaining this information is bycho osing new indeterminates y and t and

substituting into r u the following values for the variables u

y if j

i

u t if j i

j

j

t otherwise

A simple calculation shows that if the original p olynomial r factored as

Y

r u u

then after the substitution we obtain a p olynomial r which factors as

n

X Y

j i

j

A

y t r y t

j

and the least nonzero co ecientofr as a p olynomial in t is a p olynomial r

i

Y

i

r y y

i



 Chapter Parallel Resultant Computation

th

the ro ots of which are just the i co ordinates of the ane solutions of the

original system

Using subresultant techniques and the notion of primitive elements this

metho d can b e extended to provide a to ol whichiseven more useful As in

the sketchabove the following result relies on the fact that the factors of

the p olynomial in whichwe are interested are all linear Together with the

construction of generalized characteristic p olynomials presented in x it

p ermits us to reduce the problem of nding all of the zeros of n p olynomi

als in n variables to a univariate problem provided that there are only a

nite numb er of solutions altogether More imp ortantlyitprovides a wayof

representing these solutions symb olically as in the following theorem proven

indep endently by Canny and Renegar

THEOREM

Let f f bepolynomials with only a nite number of common zeros

n

We can compute a polynomial q t and rational functions r tr t

n

such that the points r r include al l of these common zeros

n

as ranges over the roots of q

The complete construction and its pro of are presented in and where

applications to problems in real geometry are also discussed The construc

tions have already proven useful in the following context when wehave found

a nite set of p oints dened by a set of multivariate equations this algorithm

allows us to reduce the multivariate problem to a univariate problem involving

just those p oints

Extensions to the Ane Case

In the ane casethe case of inhomogeneous p olynomialsthere can

not b e a purely algebraic criterion for the existence of a common solution

to sets of p olynomial equations However recent results of Kollar and

Galligo Heintz and Morgenstern do yield a parallel p olynomial time alge

braic algorithm for deciding this question The algorithm again dep ends on

obtaining go o d degree b ounds for the Nullstellensatz

According to the Nullstellensatz whenever f f R

m

V f f f f

m m

Multivariate Resultants 

m

X

g g R g f

m i i

i

The discussion of the previous sections suggest that if we can nd a degree

b ound for the p olynomials g in then we can reduce the problem of

i

determining whether is an element of this ideal to the problem of deter

mining the rank of an appropriate matrix formed from the co ecients of the

given p olynomials By reducing the problem to the problem of determining

the rank of a matrix one can then apply Mulmuleys algorithm to giveapar

allel p olynomialtime solution to the problems of deciding the solvabilityofa

set of p olynomial equations and even the problem of quantier elimination

in algebraically closed elds The essential facts are stated in the following

theorem of Kollar

THEOREM L

et f f R bepolynomials of degree at most dIfV f f

m m

then there exist polynomials g g satisfying with deg f

m i

n

deg g d for i m

i

This b ound now leads to the following algorithm for determining whether

there exist solutions to a given system of p olynomial equations

For each e let R denote the additive subgroup of R consisting

e

of all p olynomials of degree at most e Note that R is a k vector space of

e

dimension n and as a basis we can take all monic monomials of degree at

e

most e Let b e the matrix of the linear map

m

Y

n n

R R

d deg f d

i

i

m

X

g g g f

m i i

i

with rep ect to this basis and let denote the vector corresp onding to the

n

Nowwe know that the p olynomials f havea multiplicative identity R

i d

solution if and only if is in the linear span of the columns of If it is then

obtained by adding the rank of is the same as the rank of the matrix

 Chapter Parallel Resultant Computation

as a new column of If not then the ranks dier Invoking Mulmuleys

NC algorithm for determining the rank of a matrix wethus obtain an ecient

parallel algorithm for this problem

Acknowledgements Doug Ierardi was supp orted in part by NSF grant CCR

and in part by the NSF Center for Discrete Mathematics and The

oretical Computer Science DIMACS Dexter Kozen was supp orted by NSF

grant CCR the John Simon Guggenheim Foundation and the US

Army Research Oce through the ACSyAM branch of the Mathematical Sci

ences Institute of Cornell University under contract DAALC

Exercises

L

Let D b e a square matrix with entries in a graded ring S S

d

d

Supp ose that every entry of D is nonzero and homogeneous and every

minor of D is homogeneous Prove that det D is homogeneous

Let A b e a set of distinct indeterminates jAj d and let x b e another

Q

di

indeterminate Show that the co ecientofx in the p olynomial x

aA

a is a homogeneous p olynomial in k A of degree i This co ecient is called

th

the i elementary in A

Let f g be multivariate p olynomials suchthat

all irreducible factors of f are distinct ie f is squarefree

f and g are homogeneous of the same degree and

V f V g

Show that f g

Let A be the multiset of ro ots of a monic univariate p olynomial f thus

Q

f x SimilarlyletB be the multiset of ro ots of a monic

A

Exercises 

univariate p olynomial g Prove that the resultantoff and g is

Y

A

B

Show that the multivariate resultantoftwo homogeneous p olynomial s in two

variables is essentially the same as the univariate resultantoftwo p olynomi

als Explain whythisisso

Let f b e a univariate p olynomial with rational co ecients The resultantof



f and its formal f is called the of f

Calculate the discriminant of the quadratic ax bx c

Provethatf has a multiple ro ot if and only if its discriminantvanishes

Let s and t b e as in Denition and let p and q b e arbitrary

m m

p olynomials Prove that for m n

gcdp q gcds p t q s p t q

m m m m

Let f f f f b e the PRS of f and f andlet s and t be as in

n n m m

Denition Provethatfor m n

m m

f gcdf t f f s f

m m m m m

 Chapter Parallel Resultant Computation

Bibli ography

M BenOr D Kozen and J Reif The complexityofelementary algebra

and geometry JCSS

SJ Berkowitz On computing the determinant in small parallel time us

ing a small numb er of pro cessors Information Processing Letters

A Boro din J von zur Gathen and J Hop croft Fast parallel matrix

and gcd computations Information and Control

W Brown and J F Traub On Euclids algorithm and the theory of

subresultants J ACM

L Caniglia A Galligo and J Heintz Some new eectivity b ounds in

computational geometry Preprint

J Canny The Complexity of Robot Motion Planning PhD thesis MIT

J Canny Generalized characteristic p olynomials Preprint

J Canny Some algebraic and geometric problems in PSPACE In Proc

th ACM Symp Theory of Computing pages Asso c Comput

Mach May

A L Chistov Fast parallel calculation of the rank of matrices over a eld

of arbitrary characteristic In Proc Conf Foundations of Computation

Theory Lecture Notes in Computer Science pages Springer

Verlag

L CsankyFast parallel matrix inversion algorithms SIAM J Comput

 Chapter Parallel Resultant Computation

R Hartshorne Algebraic Geometry SpringerVerlag

D Ierardi The Complexity of Quantier Elimination in the Theory of

an Algebraical ly Closed PhD thesis Cornell University

J Kollar Sharp eective Nullstellensatz Preprint

D C Kozen The Design and Analysis of Algorithms SpringerVerlag

D Lazard Resolution des systems dequations algebriques Theor Com

put Sci

F S Macaulay The Algebraic Theory of Modular SystemsCambridge

Tracts in Mathematics and Mathematical Physics Cambridge U Press

H Matsumura Commutative Ring TheoryCambridge Studies in Ad

vanced Mathematics Cambridge U Press

K Mulmuley A fast parallel algorithm to compute the rank of a matrix

over an arbitrary eld Combinatorica

J Renegar On the worst case arithmetic complexityofapproximating

zeros of systems of p olynomials Technical Rep ort Scho ol of Op era

tions Research and Industrial Engineering Cornell U Ithaca New York

J Renegar A faster PSPACE algorithm for deciding the existential

theory of the reals Technical Rep ort Sc ho ol of Op erations Research

and Industrial Engineering Cornell U

J Renegar On the computational complexity and geometry of the rst

order theory of the reals Parts I I I I I I Technical rep ort Scho ol of

Op erations Research and Industrial Engineering Cornell U Ithaca New

York

J T Schwartz Fast probabilistic algorithms for verication of p olyno

mial identities J ACM Octob er

B L van der Waerden Modern Algebravolume F Ungar Publishing

Co third edition

B L van der Waerden Modern Algebravolume F Ungar Publishing Co fth edition

Exercises 

J von zur Gathen Parallel linear algebra This volume

J von zur Gathen Parallel algorithms for algebraic problems SIAM J

Comput

RE Zipp el Probabilistic algorithms for sparse p olynomials In Ng edi

tor Proc EUROSAM Lecture Notes in Computer Science pages

SpringerVerlag

 Chapter Parallel Resultant Computation

Index

algorithm ane

sets

ring

space

scheme

algebraic

extended Euclidean scheme

geometry

real

generalized characteristic p olynomial

set

basis

graded ring

Berkowitz algorithm

greatest common divisor

Hauptidealsatz

BKR algorithm

Hilb ert Basis Theorem

homogeneous

characteristic p olynomial

equations

generalized

ideal

Chistovs algorithm

Nullstellensatz

Cramers rule

p olynomial

Csankys algorithm

homogenization

hyp erplane at innity

determinant

ideal

discriminant

improp er solutions

EES

inhomogeneous

eective homogeneous Nullstellen

p olynomials

satz

sets

elementary symmetric p olynomials

matrix rank

elimination theory

Mulmuleys algorithm

Euclidean multiplicity

uresultant multivariate resultant

univariate resultant

Nullstellensatz

Zipp elSchwartz Lemma

homogeneous

eective

strong form

weak form

p olynomial

division

equations

remainder sequence

primitive element

Pro jective Dimension Theorem

pro jective space

PRS

quantier elimination

randomness

real closed eld

resolvent

resultant

matrix

multivariate

parallel algorithm for

p olynomial

system

univariate

rob otics

solid mo deling

standard emb edding

subresultant

matrix

Sylvester matrix total degree