Triangular sets for solving systems : a comparison of four methods Philippe Aubry, Marc Moreno Maza

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Philippe Aubry, Marc Moreno Maza. Triangular sets for solving polynomial systems : a comparison of four methods. [Research Report] lip6.1997.009, LIP6. 1997. ￿hal-02546252￿

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Des ensembles triangulaires p our resoudre les

systemes p olynomiaux une comparaison de quatre

methodes

Philipp e Aubry

Marc Moreno Maza

LIP Universite Paris

Place Jussieu Paris Cedex

Tel

email aubryp ossoibpfr mp ossoibpfr

resume

Quatre methodes de resolutionde systemesdequationspolynomiales sont presentees et implantees

dans un cadre commun Ces methodes sont celles de Wu Wu Lazard Laz Kalkbrener

Kal et Wang Wanb El les sont comparees sur divers exemples avec une attention par

ticuliere porteea lecacitela concision et la lisibilitedes sorties

Triangular Sets for Solving Polynomial Systems a

Comparison of Four Metho ds

Philipp e Aubry

Marc Moreno Maza

LIP Universite Paris

Place Jussieu Paris Cedex

Tel

email aubryp ossoibpfr mp ossoibpfr

January

Abstract

Four methods for solving polynomial systems by means of triangular sets are pre

sented and implemented in a unied way These methods are those of Wu Wu

Lazard Laz Kalkbrener Kal and Wang Wanb They are compared

on various examples with emphasizing on eciency conciseness and legibility of the

outputs

Introduction

In this pap er we are concerned with the following problem given a nite family F of

multivariate p olynomials over a eld k and with ordered variables X X X

n

to describ e the ane variety V F ie the common zeros of F over an algebraic closure

of k Such a description is usually given by a nite family fT T g of p olynomial

r

sets with particular prop erties a link b etween the T and F and an algorithm to

i

compute the T from F A well developed strategy since Buchbergers work Buc

i

is the following given an ordering on the monomials to choose for T the Grobner

basis of the ideal generated by F and compute it by the Buchbergers algorithm

Wu WenTsunin Wu introduced another way of solving algebraic systems which

is the one we are concerned with in this pap er In that case each T is a p olynomial set

i

such that two distinct p olynomials in T have distinct greatest variables Such a T is

i i

called a triangular set The p oints of V T where no leading co ecient of a p olynomial

i

in T viewed as univariate in its greatest variable vanishes are called the regular zeros

i

of T Then in Wus metho d the variety V F is the union of the regular zeros of

i

the T and this decomp osition can b e computed by the original Wus CHRSTREM

i

algorithm Wu This metho d has b een investigated in many pap ers Among them

Cho CG CG GM Wana Wanb Wus metho d is ecient for geometric 1

problems where the degenerate solutions are not interesting For general problems it

seems to b e dicult to obtain an ecient implementation and this metho d may pro duce

sup eruous triangular sets Wus algorithm like Buchbergers one dep ends on many

choices moreover its result is not uniquely dened

A Wus like decomp osition of ane varieties can b e obtained by Daniel Lazards

algorithm Laz But in that case the denition of triangular sets has b een strength

ened denition in order to guarantee irredundant and more canonical decomp osi

tions Our pap er rep orts a rst implementation of this metho d and shows that it can b e

ecient In MR M Moreno Maza and R Riob o o rep ort a very ecient implemen

tation of another algorithm due to Daniel Lazard Laza and called Lextriangular

This last algorithm also computes decomp ositions similar to those of Laz but the

input must b e a lexicographical Grobner basis of a zerodimensional ideal Lazards

decomp ositions have at least two interesting prop erties On one hand numeric solutions

may b e easily obtained from them b ecause Lazards triangular sets are normalized def

inition See also section in Laza for more details On the other hand Lazards

triangular sets are well suited for describing prime ideals see section in Laz

whereas there is no b ound on the minimal number of generators for a lexicographical

Grobnerbasis of a prime ideal

In Kal Michael Kalkbrener introduced another type of triangular sets called

regular chains denition together with another link b etween F and the T In that

i

case V F is the union of the closures wrt Zarisky top ology of the regular zeros

of the T Let us mention an example to see the dierence b etween Wu and Lazards

i

way of solving and Kalkbreners one We consider the system given by the following

p olynomials where the ordered variables are c s c s b a and where the

co ecients lie in the eld of rational numbers

o n

s c s c c s s c a s c c s s b c

Our implementation of Lazards algorithm Laz pro duces the decomp osition

fT T T g where

s b a b s b a b a a T b a

a c b s b a

s b a b a s b a

c b a

T a s b c b s b c c b

T a b c s s c

How this solution has to b e understo o d In T one may arbitrarily choose a and b once

ab a and obtain successively the values of the indeterminates s c s c

The triangular sets T and T describ e the case a Note that in T one may choose

arbitrarily b whereas b in T So where is the case b a It is describ ed by

T In fact if we add this equation to the initial system the computed decomp osition

is only fT g Now our implementation of Kalkbreners algorithm Kal pro duces

the decomp osition fC g where

C b a s b a b s b a b a a

a c b s b a s b c a s

s c b c a s c s b

2

In that case a p oint is a solution of the initial system if it lies in the closure of the

regular zeros of the only triangular set ab ove denoted by C Although C and T are

dierent their regular zero sets have the same closure which contains the regular zeros

of the previous T and T Note that Kalkbreners output is simpler but needs futher

computations for the zeros satisfying ab a

Triangular sets can also b e used to solve quasialgebraic systems denition In

Wanb Dongming Wang prop osed such a metho d by means of Wus triangular sets

But Wangs pro cess is dierent from Wus one and seems to b e more ecient

In the conclusion of Kal Michael Kalkbrener wrote a comparison with the algo

rithms of Ritt Wu and Lazard seems to b e interesting In the conclusion of Wanb

Dongming Wang wrote a systematic analysis and comparison among them the elim

ination metho ds of Lazard and Kalkbrener b oth theoretically and practically remain

interesting for future work The purp ose of this pap er is to compare the metho ds of Wu

Wu Lazard Laz Kalkbrener Kal and Wang Wanb In the rst

section following Lazb we introduce a coherent terminology to present their sp ec

ications Then we study some prop erties of regular chains and their connection with

towers of simple extensions denition We also lo ok into the sp ecial case of Lazard

sets In the second section we review the sp ecications of each metho d Futhermore

we give a recursive adaptation of Wangs metho d which seems to us easier to read than

the original iterative description Our implementation of Lazards metho d is based

on an algorithm for gcd computations of univariate p olynomials with co ecients in a

separable tower of simple extensions The idea is a generalisation of the one of MR

This algorithm could not take place here and will b e presented in a future pap er In

the third section we discuss exp erimentations on those four metho ds We think that a

reasonable comparative implementation should satisfy the following requirements

the corresp onding algorithms must b e implemented and run with the same hu

man material and software conditions using the same data structures and sub

routines

to make sure that each computed solution is correct

not to only fo cus on timings but also on the legibility of the ouputs and their

suitability for further uses

A strongly typed and ob jectoriented language is convenient to satisfy the rst require

ment ab ove We used the AXIOM system JS We dened

categories corresp onding to the dierent prop erties of triangular sets packages and

domains for the common subroutines Futhermore AXIOM is connected with GB

the very p owerful Grobnerengine developed by JC FaugereFau This allowed

us the nontrivial Grobnerbasis computations which are needed in order to satisfy the

second requirement ab ove

Our implementation of each metho d uses the same p olynomial domain constructor

Thus a metho d involving particular datastructures like the dynamic sets and dynamic

p olynomials Dia could not enter within our exp erimentations However we tested

each of our examples with the dynamic evaluation This metho d is only usable for

easy examples and cannot compare with the metho ds of Wu Wang Kalkbrener and

Lazard But note that the goal of dynamic evaluation is not restricted to p olynomial

system solving As we wanted to implement easily and completely each of the metho d 3

we considered we also discarded metho ds which dep end on sophisticated techniques

like Grobnerbasis computations or factorizations

In the last section we rep ort some exp erimental data on a set of test exam

ples Most of them can b e found in the data base of the europ ean research pro ject

PoSSo Com They are also available by ftp on possoibpfr in the directory

pubpapersTriangularSets Finally we investigate the computed decomp ositions

for some relevant examples and p oint out some remarks suggested by our exp erimen

tations

Triangular Sets and Towers of Simple Exten

sions

In this section we rst recall the most general denition for triangular sets de

nition This is the one used in Wus metho d Wu and in Wangs metho d

Wana Wanb Then we recall the denition of regular chains denition

which are particular triangular sets used in Kalkbreners metho d Kal Kal In

the third subsection we give a denition for towers of simple extensions denition

and we show that regular chains are suitable for enco ding every tower of simple ex

tensions Finally we study Lazard sets denition which are sp ecial regular chains

Their presentation is inspired by our adaptation of Lazards metho d Laz by means

of p olynomial gcd computations over tower of separable extensions full details will ap

p ear in Maz Before dealing with triangular sets we need some general notations

ab out rings ideals and varieties

Notations We denote by IN the set of the nonnegative integer numbers Let A b e

a ring all rings considered here are commutative no etherian rings with unit element

and E b e a subset of A We denote by hE i the ideal of A generated by E and

A

E

by AE the quotient ring of A by hE i For an element a A we denote by a

A

the residue class of a in AE If E fa a g we simply write ha a i or

l l

A

ha a i instead of hE i If E is empty we state hE i hi We denote by nzA

l

A A

the multiplicatively closed subset of nonzerodivisors of A this contains the group of

invertible elements of A and by qA the ring of fractions with numerators in A and

denominators in nzA Let I b e an ideal of A We denote by apI the asso ciated

p

prime ideals of I ie the comp onents of a minimal primary decomp osition of I

For an element h A the saturated ideal of I wrt h ie the set of the b A such

m

that there exists a p ositive integer m with h b I is denoted by I h The ideal

generated in AX by I is denoted by IX For a p olynomial p AX we denote by

I

p the image of p in qAI X obtained by mapping the co ecients of p into qAI

For a mo dule M over A and a multiplicatively closed subset S of nzA we denote by

S M the Amo dule of fractions with numerator in M and denominator in S Now

assume that A is a p olynomial ring with n variables and co ecients over a eld k Let

K b e an algebraic closure of k For an ideal I of A we denote by V I or simply

K

n

V I the ane variety of K asso ciated to I and if I ha a i we simply write

l

n

V a a instead of V ha a i Finally for W K we denote by W the

l l

closure of W wrt the Zarisky top ology over k whose closed sets are the V I for

every ideal I of A 4

Triangular Sets

Notations Let R b e an integral domain We denote by k the eld of fractions of

R Let K b e an algebraic closure of k Let n b e a p ositive integer and V a set of n

ordered variables X X X For i n let R RX X and

n i i

P kX X b e the rings of p olynomials in i variables with co ecients in R and

i i

k resp ectively We also dene R R and P k Let E R and p q R with

n n

p and q R For v V we write degp v for the degree of p with resp ect to the

variable v We denote by varE the set of the variables v V for which there exists

r E with r such that degr v If E fr g we simply write varr intsead of

varE We call the main variable of q denoted by mvarq the greatest variable of q

When E R we denote by mvarE the greatest variable of varE We call initial of

q denoted by initq the leading co ecient of q viewed as an univariate p olynomial

in mvarq We call main degree of q denoted by mdegq the degree degq mvarq

mdegq

and tail of q denoted by tailq the p olynomial q initq mvarq We denote

by algVarE the set of the variables v V for which there exists r E with r R

such that mvarr v Let v b e in V We denote by E E and E the set of the

v

v v

nonconstant p olynomials r E such that mvarr v mvarr v and mvarr v

resp ectively If E fr g we simply write E r

v v

Denition A subset T of R is called a triangular set if every polynomial of T is

n

nonconstant and if for al l p q T with p q we have mvarp mvarq

Example Let p R Let iterp b e the subset of R recursively dened as follows

n n

if p R then iterp else iterp fpg iterinitp Then iterp is a triangular

set of R whose elements are called the iterated initials of p

n

Notations Let E b e a subset of R and p q R with p and q R We write

n n

redp q if degp mvarq mdegq holds Then we write redp E if redp r

holds for every r E We write iRedp q if either iterp or red iterp q

v v

holds where v is mvarq We write normalizedp q if iterp holds where v is

v

mvarq Then we write iRedp E if iRedp r holds for every r E The same

way we dene normalizedp E We denote by premp q and p quop q the pseudo

remainder and the pseudoquotient of p by q when interpreting them as univariate in

mvarq Let T R b e a triangular set If T we dene premp T p else we

n

dene premp T prempremp T T where v is mvarT Then we denote by

v

v

premE T the subset of R whose elements are the p olynomials premr T for r in

n

E If T fq g we simply write premE q instead of premE T

Denition A triangular set T of R is called reduced resp initially reduced

n

resp normalized if for every t T denoting mvart by v we have redt T

v

resp iRedt T resp normalizedt T

v v

Example In the introduction the triangular sets T T and T are reduced and

normalized whereas C is initially reduced but neither reduced nor normalized

Notations Let p q R with q R Let S b e the multiplicatively closed subset of

n

R generated by h initp Let e b e the minimal p ower of h by which p is multiplied in

n 5

order to compute a p olynomial r by the p eudodivision algorithm such that redr q

e

and h p r hq i In many cases we have e degp mvarq mdegq and

R

n

r premp q If redp q we have e and r p Then we denote by mo dp q the

r

element of S R dened by Let T b e triangular set of R We denote by S the

e

n n

h

multiplicatively closed subset of R generated by and the initials of the elements of

n

T If p R then we write v mvarp i initp d mdegp and t tailp

p p p

Now we dene the element mo dp T of S R by iterating the following rules

n

p

T or p R mo dp T

d

p

v

mo dt T redp T mo dp T mo di T

p v p

v

0

mo dr r T

r r

d

p

T mo dv and mo di T mo dp T mo dt T

0 0

v p p

v

s s ss

r

with Thus for every p R there exist r R and s S such that mo dp T

n n

s

redr T and sp r hT i Finally we dene the element iRedp T of R by

n

R

n

iterating the following rules

a iRedp T iRedp T p

r

iRedp T iRedr T b mo dp T

v

s

d

p

c iRedi T r and mo ds i r T iRedp T iRedr v st T

p p p

v v

Thus for every p R there exist r R and s S such that iRedp T r with

n n

iRedr T and sp r hT i

R

n

Remark Note that to apply rule c in the denition of iRedp T it is necessary

to store the intermediate denominators s which app ear when applying rule b This

notion of iterated initials reduction is the weakest notion of reduction which ensures

the termination of Wus algorithm Lazb Futhermore as it limits the number of

reduction steps it leads generally to an increase of eciency in comparison with the

complete reduction ie the one based on the op eration p T mo dp T

Denition Every couple P Q where P and Q are two nite subsets of R

n

is called a quasialgebraic system in R qas for short Let P Q be a qas

n

in R The qas is called triangular if P is a triangular set of R If Q then

n n

we denote by h the product of the elements of Q otherwise we dene h We

n

call a zero of every element of the subset of K denoted by Z and dened by

Z V P n V h

The qas is called inconsistent if Z else it is called consistent The saturated

ideal of the qas is the saturated ideal of the ideal generated by P in P wrt h

n

Let T R be a triangular set We denote by T the triangular qas dened by

n

T T finitt j t T g

Then we denote by sat T the saturated ideal of T and by hT the product of

n

the initials of the elements of T Moreover every zero of T is called a regular

zero of T and ZT is also written WT and called the quasicomp onent of T

Final ly fol lowing Wana Wanb a triangular qas T Q is called ne if

V hT Z and premQ T 6

Remark Let T R b e a triangular set If T is a regular chain denition then

n

WT This will result from theorems and and prop osition The converse

is false as shown by the following example T fX X X X g Thus if

T is a normalized triangular set then WT This will result from theorem

and prop osition If WT then T is ne but the converse is false consider

T fX X X X X X X g

Let b e a qas in R To decide whether is consistent one can compute sat

n n

by means of Grobnerbases techniques GTZ CLO Lazb The answer is true

i sat P ie hT do es not lie in the radical of the ideal generated by T in

n n

P The following result shows more precisely the links b etween sat and Z

n n

Theorem Let be a qas in R Then we have

n

Z V sat

n

Let P Q b e a qas in R We denote by H the principal ideal Pro of

n

generated by h in P and by I the ideal generated by P in P It is clear that

n n

p

Z V I n V H Thus by theorem in CLO p we have Z

q

p p p

I H Finally one can check that I H I h I h V

Regular Chains

The concept of regular chains in P is introduced by Kalkbrenner in Kal The

n

denition b elow deals only with ideals and corresp onds to a particular case of system

of representations presented in Kal Let i b e a p ositive integer and I an ideal in

I

P recall that for f P we denote by f the canonical image of f in q P I X

i i i i

Denition Let i IN and T be a triangular set of R We say that T is a regular

i

chain in P and that the ideal Rep T of P is its representation if either i T

i i i

and Rep T fg or i and one of the fol lowing assertions holds

X algVarT the set T is a regular chain in P and

i i

P

Rep T ff P j P apRep T f g

i i i

is a regular chain in P for any associated prime X algVarT the set T

i i

X

i

P and we have init T ideal P of Rep T

X i

i

X

i

r

P

P

f h T i g Rep T ff P j P apRep T

X i i i

i

X

qP P X

i

i i

Remark With the notations of the ab ove denition if X algVarT then it follows

i

P

X mdeg T Thus if r P T from the condition init T P that deg

i X i X X

i i i

P

P

X r X deg T we have deg with degr X mdeg T

i i X i X

i i

Remark The following results can b e veried with general

p

p

let I an ideal in A and h A then I X I X and I h X I X h

Thus if T is a triangular set in P we have sat T hsat T i

i i i

P

i 7

Prop osition Let i be a positive integer and T be a regular chain in P

i

p

Rep i if X algVarT then Rep T h T i

i i i

P

i

ii if X algVarT then

i

m

g Rep T ff P j m IN premf T Rep T

i i X i

i

X

i

We rst assume that X algVarT Let f P and P apRep T Pro of

i i i

X

i

P

We have f i every co ecient of f viewed as univariate in X lies in P Thus

i

p p

Rep T ie f h Rep T i f Rep T i every co ecient of f lies in

i i i

P

i

Now we assume that X algVarT with t T and h initt For m IN we

i X

i

m

denote premf t by r There exists q P and IN such that

m i

m

h f q t r

m

First let us assume that f Rep T By p oint of denition there exists m IN

i

P

P

m

such that f ht i By choosing m big enough we can take the same integer m for

P

every prime ideal P in apRep T With the relation we deduce that t di

i

X

i

q

P P

i r By remark it follows that r Therefore r h Rep T vides

m m m i

X

i

P

i

Conversely assume that there exists m IN and with i we obtain r Rep T

m i

X

i

P

P

P

m

We get such that r Rep T h r and thus h f t i By denition

m i m

X

i

P P

P

m

h P therefore h is invertible It follows that f t i ie f Rep T h

i

Prop osition Let i IN and T be a nonempty triangular set of P such that X

i i

q

algVarT Let us assume that for every P ap P we have init T sat T

X i

i

X

i

Let r P such that r sat T Then we have

i i

q

sat T r degr X mdeg T

i i X

i

X

i

Pro of Then there exists IN First we assume that T Dene h init T

X

i

X

i

such that T divides h r The hypothesis on the degree implies r which proves

X

i

Q

initt by h and denote the assertion Now let us assume that T

X

i

t T

X

i

i h T Since r sat T there exists IN and q P such that hh r q T

i i X

i

X

i

P

i

q

sat Let P b e a prime ideal asso ciated to It is a classical result that h P T

i

X

i

P P

P

hh invertible Therefore T divides r As Since h P by hypothesis we have

X

i

P

P P

X we get r X deg T r and the statement follows we have deg

i i X

i

Prop osition Let i IN and T be a nonempty triangular set of P such that X

i i

q

algVarT Let us assume that for every P ap we have init T P sat T

X i

i

X

i

Let f P Then we have

i

q

q

m

sat sat T f T m IN j premf T

i i X

i

X

i 8

p

m

Pro of We rst consider f sat T Let m IN such that f sat T Then

i i

m

we clearly have premf T sat T and the result immediately follows from

X i

i

q

m

sat T prop osition Conversely let m b e an integer such that premf T

i X

i

X

i

Q

initt There exists else the result is obvious Let h We assume T

X

i

t T

X

i

m m

IN and q P such that h f q T premf T We easily obtain from

i X X

i i

q q

m m

this equality that h f hT i h Thus we have f hT i hh ie

P P

i i

p p

m

sat T It follows that f sat T f

i i

Theorem Let i IN and T be a regular chain in P Then we have

i

q

sat Rep T T

i i

For i the result is obvious Let i and let us assume that the equality Pro of

holds for i If X algVarT the equality easily follows from prop osition and

i

remark Now we assume that X algVarT From prop osition again we have

i

m

Rep T ff P j m IN premf T Rep T g

i i X i

i

X

i

q

sat we know that Rep T Since X algVar T from the previous T

i i i

X X X

i i i

transcendental case Finally we obtain the result with the prop osition

Prop osition Let i IN and T be a regular chain in P Then we have Rep T P

i i i

It follows from b oth relations of prop osition that Rep T i Pro of

i

Thus since the statement is clear for i it also holds for any i Rep T

i

X

i

Towers of Simple Extensions

From now on i f ng is a integer k A A A are rings T R

i i

is a triangular set and F is an algebra homomorphism of P into A X

i i i i

Denition The set T is a regular set of R whose asso ciated map is F and whose

i i

asso ciated tower of simple extensions is A A if one of the both assertions holds

i

i the set T is empty and F is the identitymap of P

i

is a regular set of R whose associated tower of simple i the set T

i

X

i

extensions is A A and whose associated map is denoted by F such

i i

that one of the both assertions holds

i X algVarT and we have

i

F p

i

A qA X and p P F p

i i i i i

ii X algVarT the element F init T is a unit in A and we have

i i X i

i

i hF T

i X

i

F p

i

i and p P F p A q A X hF T

i i i i i i X

i

More in cases i and ii we state F X X

i i i 9

Denition The sequence A A is called a tower of simple extensions of k

i

tose for short if there exists a regular set of R whose associated tower of simple

i

extensions is A A If T is a regular set of R whose associated tose is

i i

A A the ring A is called the topextension of T

i i

Remark Let T R b e a regular set whose asso ciated tose is A A For

i i

j i and x A note that x is either a unit in A or a zerodivisor in A More

j j j

if j i and if x is a unit in A then it is also a unit in A Prop osition gives

j j

an imp ortant example of regular sets and prop osition characterizes the zerodivisors

and units in the T s asso ciated tose

Prop osition Let T R be a normalized triangular set Then T is a regular set

i

Pro of If i the statement is clear Thus we can assume that i and that

is a regular set If X algVarT the statement is clear again If X algVarT T

i i

X

i

In order to show that init T cannot b e a we have normalizedinit T T

X X

i i

X

i

zerodivisor in T s asso ciated tose it suces to use remark together with the

X

i

following classical remark for a ring A a p olynomial p AX is a zerodivisor in

AX i there exists an element a A such that a p

Prop osition Assume that T is a regular set of R whose associated map is F and

i i

whose associated tose is A A Then for every p P we have

i i

F p mo dp T premp T p sat T

i n

F p is a unit in A i for every prime ideal P apsat T we have p P

i i n

The pro of is based on the following classical remark For an ideal I in a Pro of

I

no etherian ring A for x A the element x is a zerodivisor in AI i there exists a

prime ideal P asso ciated to I such that x b elongs to P SZ volume p

Theorem The triangular set T is a regular chain i T is a regular set

The statement results easily from prop osition and theorem Pro of

Lazard Sets

Remark In Laz Lazard introduced what we call Lazard sets denition A

Lazard set is a particular regular set whose topextension is a pro duct of elds The use

of eld pro ducts is motivated by denition and prop osition Lazard sets are built

by means of gcd computations in the sense of denition together with denition

and theorem Full details will app ear in Maz and in a future pap er

Denition Let A be a ring and p p g be in AX We say that g is

a gcd of p and p if the fol lowing holds

hp p i hg i

qAX qAX

Remark If qAX is not a principal ideal domain the p olynomials p and p do

not necessarily have gcd in the sense of the previous denition They may also have

several gcds But if their leading co ecients are not zerodivisors in A then there exist

e e nzA such that e g e g We chose this denition to generalize usual gcd

algorithms which give a Bezout relation together with a pseudodivisor see MR 10

Prop osition MR Let A A be integral domains and let A be their

i

direct product thus sums and products in A are computed componentwise Then for

every p and p in AX there exists g AX which is a gcd of p and p

Denition Let A be a ring of characteristic and p AX with positive degree

We say that p is

i primitive if the ideal of A generated by the coecients of p is the unitideal

ii squarefree if p and its derivative generate the unitideal of qAX

Prop osition Let A be a noetherian ring of characteristic and p AX with

positive degree Assume that A is a eld or a product of elds and that p is monic and

squarefree in the sense of the previous denition Then we have

i the ideal generated by p in AX is a radical ideal

ii each one of the rings qAX and AX hpi is a eld or a product of elds

proof Prop erty i is clear if A is a eld Assume now that A is a pro duct of elds

k k We denote by p the ith comp onent of p in k X k X and

n i n

by R the ideal of AX generated by p Note that the R

i i i i n i

are relatively prime ideals and that their pro duct is hpi Thus we have

AX

n

hpi R

i

AX

and prop erty i follows from the fact that the R are radical ideals Prop erty ii

i

results from the following remark of D Lazard if A is a no etherian ring where every

element is either a unit or nonnilp otent zerodivisor then A is pro duct of elds This

can b e derived from the theory of Lazard rings see Maz

Denition Let T R be a regular set The set T is called

n

i squarefree if for i n we have X algVarT F T squarefree

i i X

i

ii primitive if for i n we have if X algVarT then for j i

i

viewed as a multivariate polynomial in the coecients of the polynomial T

X

i

A X X generate the unitideal of A

j j i j

A triangular set of R is called a Lazard set if it is normalized squarefree and prim

n

itive A tose is called separable if it is associated to a squarefree regular set

Theorem Let T R be a Lazard set and let A be its topextension Then the

n

fol lowing assertions hold

i A is a product of elds

ii sat T is a radical ideal

n

iii for every p q AX there exists g AX such that g is a gcd of p and q

proof Prop erty i ii iii follow resp ectively from prop ositions and 11

Remark Let T R b e a Lazard set F its asso ciated map and A its topextension

n

Assume that A is a pro duct of m elds k k Let p q AX To compute a

m

gcd of p and q one may apply a standard algorithm in each k X But in practise the

i

k are not known So we p erform in AX the variation of subresultant gcd algorithm

i

prop osed in MR as if A was an integral domain Then we use a Dlike process

DDD to split the computations when a zerodivisor is discovered Let r R

n

with redr T To decide whether the element F r is a unit in A we pro ceed from

the following way If r the answer is false Else if r k the answer is true Else

if mvarr algVarT the answer is given by checking the invertibility of F initr

Else the answer is given by checking the invertibility of the of r and T

v

wrt v mvarr where the co ecients of those p olynomials are interpreted in the

topextension of T This pro cess is analogous to the one describ ed in MR

v

A review of the four Metho ds

In this section we rst recall the sp ecications of each metho d together with the main

prop erties of the decomp ositions that they compute A complete review of the algo

rithms could not take place here For Wus metho d one can refer to Wu or Wan

However we summarize the main features of the metho ds of Lazard and Kalkbrener

which b oth involve gcd computations over towers of simple extensions Moreover we

give a recursive presentation of the rst metho d prop osed by D Wang in Wanb

This adaptation app eared to us more concise than the original presentation

Sp ecications

Let F R b e a nite set of p olynomials The algorithms of Wu and Lazard compute

n

a nite family fT T g of initially reduced triangular sets such that

r

r

V F WT

i

i

In the case of Wus metho d one of the T say C satises the following

i

i WC V F V C

S

ii V F WC V F finitpg

pC

Such a triangular set is called a characteristic set for F Wu In the case of

Lazards metho d each T is a Lazard set Lazards decomp ositions but not Wus

i

ones are irredundant in the following sense

WT WT

j j

j

j i

Kalkbrenners metho d computes a nite family fT T g of regular chains but

r

deals rather with variety than regular zeros The decomp osition is such that

r

V Rep T V F

n i

i 12

Thus by theorems and we also have

r

WT V F

i

i

The prop osition guarantees that for every T we have WT but we may have

i i

S S

WT WT for some i

j j

j i j

Wangs metho d computes a nite family fT Q T Q g of ne triangular

r r

qas such that

r

ZT Q V F

i i

i

Such a decomp osition is pro duced by triangulerF theorem There is a no

reason for a ne triangular system pro duced by the metho d of Wang describ ed b elow

called elimination without projection in Wanb to b e necessarily consistent But

may b e due to our optimizations we never encountered inconsistent ne triangular sys

tem during our exp eriences Note that Wang prop oses also a metho d called elimination

with projection in Wanb to pro duce necessarily consistent outputs

Lazards Metho d

The main pro cedure of Lazards metho d is called intersect Given T R and p R

n n

the op eration intersectp T returns a nite family of Lazard sets fS S g such that

l

l

V p WT WS V p WT

i

Given fT T g a nite family of Lazard sets we dene intersectp fT T g

s s

as the union of the intersectp T Then given a nite subset F ff f g

i m

of R we dene intersectF T intersectf intersect intersectf T Thus

n m

intersectF pro duces a nite family of Lazard sets fS S g such that

l

r

WS V F

i

i

We will not describ e here how to pro duce irredundant decomp ositions The op eration

intersectp T pro ceeds in the following way

l If normalizedp T holds then go to step l with r p else go to next step

l If normalizedp T do es not hold compute two p olynomials q r R such that

i

normalizedr T and mo dpq r T and mo dp T mo dr T

Polynomials q and r are computed by means of an extended ie with Bezout

co ecients version of the gcd algorithm sketched in remark Here the com

putations may b e split if mo dp T is a zerodivisor The p olynomial r is also

denoted by normalizep T Now go to next step

l If r then returns fT g Else if r k then returns f g Else go to next step

l Return intersecttailr intersectinitr T and go to to next step

l Remove the content of r viewed as univariate in mvarr and go to next step

l If T fr g is a squarefree regular set then go to step l

mvarr 13

l Let v mvarr Compute a normalized wrt T gcd of r and its derivative

v

wrt v while interpreting their co ecients in the topextension of T here

v

computations may b e split Let g b e this gcd replace r by p quor g Thus

T fr g is now a squarefree regular set Go to step l

v

l Let v mvarr Dene T ft t g with mvart mvart

k l k l

v

Compute D intersectt intersect intersectt T fr g Then remove

l k

v

from D any triangular set U such that normalizeinitt U for some

i

mvart

i

i fk l g Now go to next and last step

l return intersectp D where p is the input p olynomial

Kalkbreners Metho d

Kalkbreners Metho d is not so incremental as Lazards one We think that a go o d way

to sketch this metho d is to give the algorithm of decomp osition with the sp ecications

of Kalkbreners algorithm for computing gcd over towers of extensions Kal

algorithm gcd C F

n

Input C a regular chain in P and F a nite subset of R

n n

Output fC g C g g where every C is a regular chain in P and every

s s k n

g is a p olynomial in R such that

k n

S

s

apRep C apRep C

n k n

for all P apRep C

n k

F s and g

P

P

g is the gcd of F in q P P X for each k F

k n n

if g k and mvarg X then initg P

k k n k

if g k and mvarg X then g P

k k n k

g hRep C F i

k n k

P

n

algorithm decomp ose F

n

Input F a nite subset of R

n

q

p

Output regular chains T T of P such that hF i sat T

i r n n i

P

n

decomp ose F

n

F F n fg

empty F fg

F R f g

F F R

n

decomp ose F

n

for C rep eat

gcd C F n F

n

for C g rep eat

i i

g fC g

i i

mvarg X decomp ose F fg g

i n n i

fC fg gg decomp ose F initg

i i n i

return 14

Wangs Metho d

Let P Q a qas in R such that mvarP X The algorithm eliminer

n i

presented b elow prop osition splits the qas into several qas which contain

at most one equation with X as main variable see denition Its pro of is based

i

on the following lemma Wana and lemma which is a practical remark whose

pro of is left to the reader

Denition Let i n and P Q a qas in R such that P R and

n i

Q R We call elimination of the variable X in a set of triplets P Q

n i k k k

such that for any k P Q and are nite subsets of R and verify the fol lowing

k k k n

conditions

i P mvarP X

k k i

ii t R n R j ftg

k i i k

S

ZP f g Q iii ZP Q

j j j

P Q

j j j

Lemma Let f a non constant polynomial in R and P Q a qas in R Then

n n

ZP ff g Q ZpremP f ff g Q finitf g ZP finitf tailf g Q

Lemma Let P Q be a qas in R and f R n R Then

n n

initf Q ZP ff g Q ZP ff g premQ f

Prop osition Let v be a variable in V and P Q a qas in R such that mvarP

n

v Then the algorithm eliminerv P Q below computes an elimination of the variable

v in the qas P Q In particular if the output of the algoritm is the empty set then

ZP Q

eliminerv P Q

P P n fg

Q or P R returnf g

P return fP Q g

v

f a p olynomial in P with minimal degree in v

v

P P n ff g finitf tailf g P

v

v

Q Q finitf g

empty P n ff g return f P premQ f ff gg eliminerv P Q

v

v

P prem P n ff g f ff g P

v

v

return eliminerv P Q eliminerv P Q

Pro of We will prove termination and correctness by induction on sP

P

degp v

pP nfg

If a constant o ccurs in P or Q the result is obvious Else if sP then P

v

and the algorithm terminates The correctness is obvious Now we assume that

sP ie P is not empty First we remark that sP sP since deginitf v

v

and degtailf v degf v Then two cases can b e distinguished

P ff g By induction eliminerv P Q terminates and is correct Therefore

v

eliminerv P Q terminates The correction follows from application of lemma

and lemma 15

P n ff g Let us denote P n ff g by P For any p P we have

v v

degpremp f v degf v degp v Since P is not empty we thus ob

tain spremP f sP and consequently sP sP Then termination

and correctness follow by application of lemma and induction hypothesis

By decreasing use of the algorithm eliminer we easily obtain a triangulation of any

qas as we can see now with the following algorithm

Theorem Let i n and P Q a qas in R such that P R Let T a trian

n i

gular set of R such that T R Then the fol lowing algorithm triangulerP Q T

n i

computes a nite family fT Q T Q g of triangular qas such that

r r

r

ZT Q ZP T Q

k k

k

triangulerP Q T

P P n fg

Q or P R return f g

empty P return fT Qg

v mvarP

eliminerv P Q

S

triangulerP Q T return

j j j

P Q

j j j

Pro of The pro of of the algorithm is obtained by induction on the smallest integer

such that P R which we will denote by iP For iP ie P R the result is

i

obvious Now assume that iP We can eliminate the cases Q P R and

which terminate immediately and are clearly correct Then by sp ecications of

the algorithm eliminer we obtain

ZP T Q ZP Q V T ZP f g T Q

K j j j

P Q

j j j

Now we state T f g T The triplets P Q T satisfy the input conditions of

j j j j j

trianguler And since iP iP the result follows from induction

j

Implementation

General Requirements

In the introduction we gave three requirements in order to make a fair comparison

of the metho ds for p olynomial system solving The most imp ortant is to implement

and run the corresp onding algorithms with the same human material and software

conditions More generally given a system of equations to b e solved we want that

the dierence b etween the corresp onding computations only dep end on the dierence

b etween the corresp onding algorithms In particular we want our implementations of

those metho ds to use the same data structures and subroutines We will describ e this

last p oint b elow

Another imp ortant requirement is to make sure that each computed solution by

one of the implementations of the four metho ds is correct We concentrated on this 16

last p oint instead of the search of very optimized implementations We think that only

checking by hand some computations necessarily simple pro duced by an implemen

tation is not sucient to make sure that this implementation is correct esp ecially for

mixed dimensional problems We had a wrong implementation of Wus metho d during

three years solving Lius example in sec due to a programming mistake in the

management of the elimination of the redundant branches Thanks to our checking

pro cess to b e describ ed b elow we discovered this bug However our current imple

mentation of Wus metho d do es not solve Lius example any more

This checking pro cess

has b een intensively tested for more than one year

is based on simple and well known algorithms and

is implemented in a direct way in AXIOM as an overlevel of the GB software

Fau

Thus it will b e considered as certainly reliable

In our analysis of the computed solutions we also lo ok for other informations than

timings or correctness Given a solution we want to know if some of the computed

triangular sets are inconsistent or if some quasicomp onents are contained in another

quasicomp onent or in the closure of another quasicomp onent This could also b e

done as we will see

Description of the implementation

Each implementation of the four metho ds uses the same AXIOM domain for p olynomi

als with a sparse and recursive representation We rst dened an AXIOM category

for nite subsets of R This category exp orts and implements op erations on sets ide

n

als and varieties like I J I J and I p I p where I J R denote ideals

n

and p R is a p olynomial We implement these op erations by means of Grobnerbases

n

techniques CLO in an AXIOM package using the connection b etween AXIOM

and GB the very p owerful Grobnerengine developed by JC FaugereFau Then

we wrote an AXIOM category for general triangular sets of R This category ex

n

p orts and implements basic op erations like T v T and p T premp T

v

and p T iRedp T notation and notation where v is a variable and T R

n

is a triangular set It also exp orts and implements more sophisticated op erations like

T sat T in order to check consistency of a triangular set

n

F R fT T R g V F WT in order to check the

n r n i i

correctness of a computed decomp osition

Moreover this category exp orts but do es not implement an op eration F R

n

zeroSetSplitF which represents any metho d for solving p olynomial system by means

of triangular sets From the category of general triangular sets we derived a category

for towers of simple extensions It exp orts the asso ciated map of a tose implemented

with the op eration p T mo dp T notation It also exp orts op erations like

p T ismo dp T aunit Finally from the category of tose we derived three

categories corresp onding to particular prop erties of regular sets 17

a category for the regular sets T R such that algVarT fX X X g

n n

called algebraic tose

a category for the normalized regular sets called normalized tose and

a category for the squarefree regular sets called separable tose

Now each metho d is an implementation of the op eration F R

n

zeroSetSplitF in an AXIOM domain of the suitable category For instance Kalk

breners metho d is implemented in an domain which b elongs to the tose category

and which is called RegularChains see the picture b elow Note that we implemented

the lexTriangular algorithm MR in an AXIOM domain called LexTriangular and

which b elongs to b oth categories of normalized tose and algebraic tose

PolynomialSetCategory

WangTriSet TriangularSetCategory WuTriSet

TowerOfSimpleExtensions (TOSE) RegularChain

AlgebraicTOSE NormalizedTOSE SeparableTOSE

LexTriangular NormalizedAlgebraicTOSE NormalizedSeparableTOSE LazardTriSet

Examples

We now present two tables of results of our exp eriments They are resp ectively ded

icated to dimension zero and p ositive dimension We give b elow the sources of our

examples For every example F and every metho d which decomp oses V F into tri

angular systems we give two informations The rst one is the computing

r

time evaluation and garbage collector The second one

is n n where n denotes the number of solutions of if V F has

r i i

dimension

else d d where d denote the dimension of sat

r i n i

In order to make more concise these sequences of numbers we use some notations

Let us take the example with Wangs metho d in the rst table The sequence

means that the decomp osition contains two triangular sets with solutions

one triangular set with solutions and two triangular sets with solutions The same

kind of notation applies for sequences of dimensions Futhermore in that case we

precise the inclusions b etween the saturated ideals of the comp onents when these

inclusions could b e computed 18

dim Exp Wang Wu Kalkbrener Lazard

2 2

3 3

2 5 4 5 4

2 2 2

2

5 5

p osdim exp Wang Wu Kalkbrener Lazard

2

a a a a a

b b b b b

2 2

3 3 3

2 2 2

2 2 2

3 4 3 2

3 3 5 2 3 3

3 2 3 4

5

4 4 4 12 14 5 2 3

a

2 4 2 8 3 2 3

2 8 5 12 13 5 7 5

3 4

2

19

Ex Source or description

Solotare Com

Mo eller Com

Trinks BGK

Trinks BGK

Katsura BGK

2 2 2

system L fx x x x x x x x x g

2 2 3 1 3 1 2

1 2 3

with x x x

1 2 3

3 3 3

x x x x x x x x x x x system L fx

4 3 4 1 2 2 3 4 1 3

3 2 1

3

x x x x g with x x

1 2 3 1 4

4

2

system R fx x x x x x p p p g

5 1 1 2 1 2 3 4 5

2

i1

i

where p x x x x x and x x

i i i1 i 1 5

i1 i

2

x x x p p p p g system R fx x x

2 1 2 3 4 5 6 6 1 1

2

with x x

1 6

2

system R fx x x x x x p p p p p g

7 1 1 2 1 2 3 4 5 6 7

2

with x x

1 7

Caprasse Com

f f

2 2 2

Singular Points F ff g where f y xy x y x

x y

Cyclic Laza

Discriminant degree ex in Wan

Cyclic Laza

Buchberger Com

DonatiTraverso Wanb

Alonso Com

Rob ot Plano facilsee introduction

Euler theorem Dia

Wang Wana

Wu Wana

2 2 2 2

Rob ot Plano dificil F fs c s c l c l s l s c b

3 2 2 1 3 2 1

1 1 2 2

l s s l c l c ag with b a l l c s c s

3 2 1 3 2 2 1 3 2 2 2 1 1

Butcher BGK

2 2

Rob ot Romin F fds a dc b l c l c d s l s c s c

1 1 2 2 3 3 2 3 3

1 1

2 2 2 2

s c s c g with d c b a l l c s c s c s

3 2 3 3 2 2 1 1

2 2 3 3

f FdSMR

Neural Network Kal

Liu Liu

Let us examinate the outputs of two examples from the tables Alonso example

corresp onds to a prime ideal of dimension whose Kalkbreners metho d describ es with

only one regular chain C The other algorithms extract p oints which are in the closure

of the regular zeros of C and provide similar results We eectively veried in these

WC cases that b oth the outputs of dimension where contained in

Wangs metho d

t t fv tr tuz r t u v z t t x t u t t

r y t u u r x u t u r t u r r t

u t v t t t tz t y t t t t t

Wus metho d

r t u r v tr t tz t y t u

r t u v tr tz t t x t u t t

v tr tuz r y t u u r x u t

Kalkbreners metho d

v tr tuz r y t u u r x u t

Lazards metho d

r t u t v t t t tz t y t t

v tr tuz r y t u u r x u t

r t u v z t t x t t 20

The following example Singular p oints on a curve example of dimension zero

is slightly dierent This example seems not to o dicult but Wus metho d failed

Here the metho ds give dierent results and we can note that Lazards decomp osition

is ecient for timing and legibili ty

Wangs metho d

fx x xy x g x x x x

x x x x x y x x x x x

Kalkbreners metho d

x x x x x y x x

x x x y x fx y g

Lazards metho d

fx x y xg fx y xg fx y g fx y g

Conclusions

For easy examples we remark that all metho ds generally have go o d computing times

and that the legibility of the outputs they pro duce is satisfactory Nevertheless Wus

metho d failed in some rather easy zerodimensionnal examples namely Caprasse R

Futhermore for b oth cases of dimension and p ositive dimension Wus metho d solves

clearly less problems than the other metho ds And for the most dicult examples Wus

metho d can solve the outputs are hard to read see Rob ot Romin In our opinion

the reason is the following Wus metho d cannot split the computations in order to

obtain several triangular sets b efore computing a characteristic set of F which is

sometimes hard to compute esp ecially for zerodimensionnal problems whereas the

other metho ds may split their computations earlier More generally it seems that

metho ds based on gcd computations over tower of simple extensions namely those of

Lazard and Kalkbrener may discover factorizations that other metho ds cannot nd

Cyclic

Let us now concentrate on Wangs metho d This metho d may b e very ecient for

dicult examples But the pro duced outputs are generally less legible than the ones

of Kalkbrener and Lazard Futhermore as Wus metho d the metho d of Wang is disap

p ointing in not to o dicult zerodimensional examples namely Caprasse and Cyclic

Note that whereas our implementation of Wus metho d pro duced some inconsistent

triangular sets this never happ ened with our implementation of Wangs metho d

Kalkbreners metho d is the only metho d which solves every example except Liu and

often pro duces the most concise outputs except for Cyclic Futhermore this metho d

has the b est timings for dicult problems lile f Rob ot Romin Neural Network But

one has to keep in mind that this metho d solves p olynomial systems in a more lazy

way than the other three This metho d is also inecient for some zerodimensional

examples Cyclic R whereas Lazards metho d succeeds with these examples The

reason seems to b e the use of normalization in Lazards metho d which can replace

big algebraic expressions by a single integer number in zerodimensional examples

However normalization and squarefree factorization over tower of separable ex

tensions are time consuming This is the reason why Lazards metho d may also b e

inecient in some not dicult examples Katsura Trinks dicult For describing 21

ane varieties by means of regular zeros of triangular sets Lazards metho d gives

the b est outputs Moreover this is the only metho d which pro duces irredundant de

comp ositions We think that the metho ds based on computation of gcd over tower of

extensions are promising The exp eriments show that they must b e further investigated

for more eciency A future challenge consists in using our practice of the algorithms

of Kalkbrener and Lazard to take advantage of b oth metho ds and resolve more dicult

problems

References

BGK W Bo ege R Gebauer and H Kredel Some examples for solving systems

of algebraic equations by calculating gro ebner bases J Symb Comp

Buc B Buchberger Ein Algorithmus zum Aunden der Basiselemente des

Restklassenringes nach einem nul ldimensionalen Polynomideal PhD the

sis Innsbruck

CG SC Chou and XS Gao RittWus decomp osition algorithm and geom

etry theorem proving In Proc CADE pages Kaiserslautern

Germany

CG SC Chou and XS Gao Solving parametric algebraic systems In Proc

ISAAC pages Berkeley California

Cho SC Chou Mechanical Geometry Theorem Proving D Reidel Publ

Comp Dordrecht

CLO D Cox J Little and D OShea Ideals Varieties and Algorithms

SpingerVerlag

Com Europ ean Commission PoSSo Polynomial System Solving Research

Project Esprit Scheme Pro ject No Has b een extended to

the FRISCO pro ject

DDD J Della Dora C Discrescenzo and D Duval Ab out a new metho d

metho d for computing in algebraic number elds In Proc EUROCAL

Vol volume of Lect Notes in Comp Sci pages Springer

Verlag

Dia Teresa Gomez Diaz Quelques applications de levaluationdynamique Uni

versitede Limoges Thesede Do ctorat

Fau JC Faugere Resolutiondes systemesdequationsalgebriques Universite

Paris Thesede Do ctorat

FdSMR JC Faugere F Moreau de Saint Martin and F Rouiller Design of

nonseparable bidimensional wavelets and lter banks using Grobnerbases

techniques IEEE Trans in Signal Processing preprint

GM G Gallo and B Mishra Ecient algorithms and b ounds for WuRitt

characteristic sets In Proc MEGA pages

GTZ P Gianni B Trager and G Zacharias Grobner Bases and Primary

Decomp osition Of Polynomial Ideals J Symb Comp 22

JS Richard D Jenks and Rob ert S Stutor AXIOM The Scientic Compu

tation System SpringerVerlag AXIOM is a trade mark of NAG

Ltd Oxford UK

Kal M Kalkbrener Three contributions to elimination theory PhD thesis

Johannes Kepler University Linz

Kal M Kalkbrener A generalized euclidean algorithm for computing trian

gular representations of algebraic varieties J Symb Comp

Kal M Kalkbrener Algorithmic prop erties of p olynomial rings Masters

thesis Dep of math Swiss Federal Institute of TechnologyZurich

Laz D Lazard A new metho d for solving algebraic systems of p ositive dimen

sion Discr App Math

Laza D Lazard Solving zerodimensional algebraic systems J Symb Comp

Lazb D Lazard Terminology for triangular and characteristic sets Technical

rep ort UniversiteParis PoSSo rep ort

Liu ZJ Liu An algorithm on nding all isolated zeros of p olynomial equa

tions MM Research Preprints

Maz M Moreno Maza Calculs de Pgcd audessus des Tours dExtensions Sim

ples et Resolutiondes SystemesdEquations Algebriques PhD thesis Uni

versiteParis Preprint

MR M Moreno Maza and R Riob o o Polynomial gcd computations over tow

ers of algebraic extensions In Proc AAECC pages Springer

SZ P Samuel and O Zariski Commutative algebra D Van Nostrand Com

pany INC

Wan D M Wang On Wus metho d for solving systems of algebraic equa

tions Technical Rep ort RISCLINZ Series no Johannes Kepler

University Austria

Wana D M Wang An implementation of the characteristic set metho d in Maple

In Proc DISCO Bath England

Wanb D M Wang Some improvements on Wus metho d for solving systems of

algebraic equations In Wu WenTsunand Cheng MinDe editors Proc of

the Int Workshop on Math Mechanisation Beijing China Institute

of Systems Science Academia Sinica

Wana D M Wang An elimination metho d based on Siedenbergs theory and

its applications In F Eysette and A Galligo editors Comptutational

Algebraic Geometry pages BirkhauserBoston Inc

Wanb D M Wang An elimination metho d for p olynomial systems J Symb

Comp

Wu W T Wu A zero structure theorem for p olynomial equations solving

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