Computing representations for radicals of finitely generated differential ideals

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generated differential ideals

François Boulier, Daniel Lazard, François Ollivier, Michel Petitot

To cite this version:

François Boulier, Daniel Lazard, François Ollivier, Michel Petitot. Computing representations for radicals of finitely generated differential ideals. 1999. �hal-00139061�

Computing representations for radials of nitely

generated dierential ideals

FranoisBoulier y

,DanielLazard,FranoisOllivierandMihelPetitot

UniversiteLilleI,LIFL,59655 Villeneuved'Asq CEDEX,Frane

UniversiteParisVI,LIP6,75252 ParisCEDEX05,Frane

EolePolytehnique,GAGE, 91128Palaiseau CEDEX,Frane

UniversiteLilleI,LIFL,59655 Villeneuved'Asq CEDEX,Frane

boulierli.fr,lazardposso.lip6.fr,olliviergage.polytehnique.fr,petitotli.fr

TehnialreportIT306oftheLIFL.Unpublished.

(ReeivedJuly1997(revisedFebruary 1999))

Thispaperdeals withsystems ofpolynomial dierential equations, ordinaryor with

partialderivatives.TheembeddingtheoryisthedierentialalgebraofRittandKolhin.

Wedesribeanalgorithm,namedRosenfeld{Grobner,whihomputesarepresentation

fortheradialpofthedierentialidealgeneratedbyanysuhsystem.Theomputed

representationonstitutesanormalsimplierfortheequivalenerelationmodulop(it

permits to test membership in p). It permits also to ompute Taylor expansions of

solutionsof.Thealgorithmisimplementedwithinapakage y

inMAPLEV.

Introdution

Thefollowingsystem(whihhasnophysialsigniane)isasystemofthreepoly-

nomialdierentialequationswith partialderivatives.

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

x u(x;y)

2

4u(x;y)=0;

2

xy

u(x;y)

y v(x;y)

u(x;y)+1=0;

2

x 2

v(x;y)

x

u(x;y)=0:

Inthefollowing,wedenote(forshort) derivationsusingindies.Thesystembeomes

8

<

: u

2

x

4u=0;

u

xy v

y

u+1=0;

v

xx u

x

=0:

y

Apartofthiswork(inpartiularthe MAPLEpakage) wasrealizedwhiletherstauthorwasa

postdotoralfellowattheSymboliComputationGroupoftheUniversityofWaterloo,N2L3G6Ontario,

Canada.

y

ThepakageisavailableforMAPLEVR3andR4.ItisgoingtoenterthemainlibraryofMAPLE

VR5.

TheRosenfeld{Grobneralgorithmthatwepresentin thispaperomputesarepresenta-

tionoftheradialpofthedierentialideal z

generatedby.Thisrepresentationtellsus

inpartiularthatthesolutionsof(whihturnouttobepolynomials)dependonthree

arbitraryonstantsandpermitsustoomputeTaylorexpansionsofthesesolutions.Ifwe

expandthemin theneighborhoodoftheoriginthenthearbitraryonstantsareu(0;0),

v(0;0)and v

x

(0;0). Foru(0;0)=5, v(0;0) =421and v

x

(0;0)= ouralgorithm gives

us(omputationsaredetailedin setion8)

u(x;y)=5+x p

10 p

2+y p

10+x 2

+xy p

2+ 1

2 y

2

;

v(x; y)=421+x+2y p

2+ 1

2 x

2 p

10 p

2+xy p

10+

1

4 y

2 p

10 p

2+ 1

3 x

3

+ 1

2 x

2

y p

2+ 1

2 xy

2

+ 1

12 y

3 p

2:

Theappliedmathematialtheoryisalled dierential algebra.Itwasinitiatedmostly

byFrenhandAmerianresearhersattheearlytwentiethentury(Riquier(1910),Janet

(1920)and(1929)andRitt(1932))andreallydevelopedbytheAmerianteamsofRitt

(1950) and Kolhin (1973). Dierential algebraaims at studying dierential equations

fromapurelyalgebraipointofview.Itismuhlosertoordinaryommutativealgebra

thantoanalysis.

TheRosenfeld{Grobneralgorithm representstheradialof thedierentialidealgen-

eratedbyanynitesystemofpolynomialdierentialequationsasaniteintersetion

ofdierentialidealsr's(thatweall regular).

p= p

[℄=r

1

\\r

n :

Eah regular dierentialideal r

i

is presented by a set of dierential polynomialequa-

tionsC

i

whihsatises:

1 C

i

isaanonialrepresentativeofr

i ,

2 C

i

redues tozeroadierentialpolynomialpifandonlyifp2r

i .

Therefore,thesetof theC'sonstitutesanormalsimplierfortheequivalenerelation

modulop(i.e.analgorithmwhihdeidesmembershipinp).Thesimplierisnotanon-

ialfortherepresentation mayontainredundantomponents:everydierentialprime

idealwhihisminimaloverpisminimaloveratleastoneoftheregulardierentialideals

produedbut theonverseisnottrue.

Assumethesolutionsofpdependonnitelymanyarbitraryonstants.Thealgorithm

separatesthesolutionswhihdonotdependonthesamenumberofarbitraryonstants.

Inourintrodutoryexample,onlyoneregulardierentialidealwasprodued.Thisproves

thatallthesolutionsofpdependonthree arbitraryonstants.

AnimplementationofthisalgorithmwasrealizedfortheMAPLEVomputeralgebra

software.Itisembedded inapakagenameddiffalg.

z

We make preise in further setions some of the notations and the terminology used in this

Usedtheorems

TheRosenfeld{Grobneralgorithmrelies mainlyonthreetheorems:

1 a theorem of zeros (Hilbert's Nullstellensatz), whih states that a polynomial p

belongstotheradialofanidealpresentedbyabasisifandonlyifeverysolution

ofisasolutionofp;weapplythistheoreminthealgebraiandinthedierential

ase,

2 alemmaofRosenfeld, whih givesasuÆientonditionso thatasystemof poly-

nomial dierential equations admits a solution if and only if this same system,

onsideredasapurelyalgebraisystemadmitsasolution,

3 alemmaofLazard,whihestablishes that eah regularidealrisradial andthat

allitsprimeomponentshaveasameparametriset(thispropertyisstrongerthan

\deninganunmixedalgebraivariety").

It utilizesonlytheoperationsand theequalitytest with zeroin thebase eldof the

equations: we referto Ritt's redutionalgorithms, omputations of Grobner bases and

splittingssimilartothoseintheeliminationmethodsofSeidenberg(1956).Inpartiular,

itdoesnotneedanyfatorization.

New results

TheRosenfeld{GrobneralgorithmwasrstdesribedbyBoulier(1994)andimproved

byBoulieretal.(1995).Thispaperontainsnewresults.

Wegiveinsetion2aproofofLazard'slemmawhihismorepreisethantheonewe

gavein Boulieretal.(1995),lemma2,page161.

WegiveanoriginalpresentationofthefundamentalRosenfeld'slemma.Wepresentit

asapropertyofsomelassofsystemsofpolynomialdierentialequationsandinequations

insteadofapropertyofsomelassof setsofdierentialpolynomials.

WegiveaversionofRosenfeld'slemma moregeneralthanthe oneof Rosenfeld(this

was alreadyproven byBoulier (1997))and notontainedin Kolhin's version.Briey,

ourversiononlyimposestotheidealstobesaturatedbytheseparantsofthedierential

polynomials(andnomorebytheinitials).Italsoonlyimposestothesetofequationsto

be triangularinsteadofautoredued(but thisis anedoti).Sine Lazard'slemma also

holdsin suhasituation,weformulateourtheorems withoutonsideringtheinitials of

dierentialpolynomials(though we doitin our implementation foreÆieny reasons).

Thisisanimprovementw.r.t.Kolhin'stheory.

Weprovenewresultsforregularideals:theorems 4.2and6.1. Theformerpermits to

omputethe minimaldierentialprime omponentsof regularidealsand provides also

informationsabouttheseprimeidealswithouthavingtoomputethem;thelattergives

usanoriginalpresentationofawellknownproofaboutformalpowerseries.

The algorithm presentedis muh moreeÆient than the one of 1995. It applies for

polynomial dierentialequations an analogue of the seond riterion proven by Buh-

berger(1979)forGrobnerbases.Ourimplementationofthisriterionwasdesignedafter

Comparison with othermethods

There is a strong relationshipbetween our algorithm and Seidenberg's work.Anton

Seidenberg (1956) designed elimination algorithms for systems of ODE and PDE in

harateristi zeroand non zero.His PDEelimination algorithm in harateristi zero

atually solves the same problem we are solving: deiding membership in the radial

ofanitelygenerated dierentialideal.Heproved(theorem6,page51)ananalogueof

Rosenfeld'slemmawhihisabitweaker(restritiontoorderlyrankingsonthederivatives

ofasingledierentialindeterminate)andmoretehnial(noteRosenfeld(1959)presents

hislemmaasanewversionofSeidenberg'stheorem).Inhistheorem11,page59heshows

that,ifisasystemwhihsatisesthehypothesesofhistheorem6theneveryalgebrai

solutionoffurnishesauniquedierentialsolution.Heshowedlater(Seidenberg,1969)

howdierentialsolutionsanbeonvertedasformalpowerseries.

There are dierenesbetween Seidenberg's algorithm and ours.The mostimportant

isthe following:the Rosenfeld{Grobneralgorithmomputesarepresentationof radial

dierentialidealswhihanbeusedafterwardsfortestingmembershipintheidealmany

timesafterwardswhileSeidenberg'sdeidesifadierentialpolynomialpbelongs tothe

radialofthedierentialidealgeneratedbyanitefamilybyeliminatingsuessively

allthedierentialindeterminateswhihourinthesystem=0; p6=0inordertotest

ifthissystemadmitssolutions(Hilbert'stheoremofzeros).Theanswerofhisalgorithm

isaboolean.

Another important dierene: Seidenberg's elimination algorithms are restrited to

elimination rankings between dierential indeterminates whih indue very explosive

omputations, whileorderly rankingsare handled bytheRosenfeld{Grobneralgorithm

(thisistheaseforinstanein ourintrodutoryexample).

Ritt(1950)gaveamethod to deompose theradial ofan ordinarydierentialideal

as an intersetion of prime dierential ideals, providing a harateristi set for eah

of these ideals. This deomposition is notthe minimal one beause of the redundany

problem(stillopen).Thatalgorithmisinonvenientbeauseitisonlypartiallyeetive:

it proeeds by fatorization over a tower of algebrai eld extensions of the eld of

oeÆients.Toourknowledge,ithasnotbeenimplemented.ItonlyappliesforODE.

Wu Wen Tsun(1987)designed avariantof Ritt'salgorithm for ordinarydierential

equations,withanotionofharateristisetweakerthanRitt's(e.g.aharateristiset

in thesenseof Wumayhavenosolution).Otherauthors (e.g.Wang (1994))developed

laterWu'sandSeidenberg'sideas.These algorithmsonlyapplyforODE.

Ollivier(1990)andCarra-Ferro(1987)haveindependentlytriedtogeneralizeGrobner

basestosystemsofordinarypolynomialdierentialequations.ThesedierentialGrobner

basesareingeneralinnite,evenforODEsystems.

Anotherdenition of dierentialGrobnerbases wasattempted byManseld (1991).

The algorithm DIFFGBASIS, implemented in MAPLE, utilizesRitt's algorithm of re-

dution and then always terminates. It handles PDE systems. In general however, it

annot guarantee itsoutput to beadierential Grobner basis.Note that themember-

shipprobleminanarbitrarydierentialidealisundeidable(Galloetal.,1991),andthe

membershipproblemofanitelygenerateddierentialidealisstillopen.

Bouzianeetal.(1996)andMa^arouf(1996)designedreentlyavariantoftheRosenfeld{

Grobneralgorithm.TheystartedfromthealgorithmofKalkbrener(1993)whihompute

amethodforomputingharateristisetsofprimedierentialidealsdierentfromour

methodsgiveninOllivier(1990),Boulier(1994)andBoulieretal.(1995),setion5,page

164.

Reid et al. (1994)and Reid et al. (1996)developed algorithms for studying systems

of PDEand omputing Taylorexpansionsof their solutions.These methods are based

moreondierentialgeometrythanonalgebra.Theydonotlaimtobeasgeneralasthe

Rosenfeld{Grobneralgorithm.

Organization of the paper

Setions1and2dealwithommutativealgebra.Theformerontainspreliminaries;in

thelatter,weproveLazard'slemmaandshowhowsomeomputationsanbeperformed

in dimensionzero. Setion 3ontainsdierentialalgebrapreliminaries. Insetion 4we

proveourversionofRosenfeld'slemmaandsometehnialresultswhihwillbeusedfor

eÆiently testingthe oherenehypothesis ofthis lemma (inpartiular, weshow there

ouranalogueofBuhberger'sseondriterion).Setion 5showshowtorepresentsradi-

aldierentialidealsasintersetionsofregulardierentialideals.Thisistheoreofthe

Rosenfeld{Grobneralgorithm. Inthe nextsetion, weshowhowto omputeanonial

representativesforregular dierentialideals andwestatethe Rosenfeld{Grobneralgo-

rithmasatheorem(theorem6.4)with aneetiveproof.Thealgorithmisobtainedby

translating theproof in any programminglanguage.In setion 7we explain how alge-

braisolutionsof regular dierentialideals anbeexpanded asformal powerseries. A

fewexamplesaredevelopedinthelastsetion.

1. Commutative algebrapreliminaries

LetR=K[X℄beapolynomialringwhereKisaeldandX isanalphabet (possibly

innite) endowedwith anordering R.Let p2RnK beapolynomial.The leader ofp

isthegreatestindeterminatex2X w.r.t.Rwhihappearsin p.Itisdenoted ldp.Let

d = deg (p;x) be the degree of p in x. The initial i

p

of p is the oeÆientof x d

in p.

Theseparant s

p

of pis thepolynomialp=x.Therankof pis themonomial x d

. It is

denoted rankp. Therankof aset of polynomialsis theset ofranksof the elements of

theset.

IfARnK isasetofpolynomialsthenI

A

(respetivelyS

A

)denotestheset ofthe

initials(respetivelyseparants)oftheelementsofAandH

A

=I

A [S

A .

If pand q are two polynomials with ranksx d

and y e

then q < pif y <x ory =x

ande<d.

Let A = fp

1

;:::;p

n

g and A 0

= fp 0

1

;:::;p 0

n 0

g be two nonempty subsets of RnK.

Renamingthepolynomialsifneeded,assumerankp

i

rankp

i+1

andrankp 0

j

rankp 0

j+1

foralli<n,j <n 0

.ThesetAissaidtobelessthanA 0

ifthereexistssomeimin (n;n 0

)

suhthatp

i

<p 0

i

andrankp

j

=rankp 0

j

for1j<ielseifn>n 0

andrankp

j

=rankp 0

j

for 1j n 0

. Two sets ofpolynomialssuh that none ofthem is lessthan the other

onearesaidtohavethesamerank.

AsubsetAofRnK issaidtobetriangular iftheleadersofitselementsarepairwise

dierent.

IfAR then (A)denotesthesmallestidealof RontainingA.If ais anidealofR

thentheradial p

aofaistheidealofalltheelementsofR ,apowerofwhihliesina.

ringR =K[X℄(X nite)is a nite intersetion of prime ideals whih is unique when

minimal.

Aomponent(sayp

1

)ofanintersetionr=p

1

\\p

n

issaidtoberedundant w.r.t.r

ifr=p

2

\\p

n

.Anelementpof aringR issaidtobeadivisor of zeroifp6=0and

thereexistsin Ranelementq6=0suhthat theprodutpq=0.

Ifr isanidealand S is anitesubset ofaringR then thesaturation r :

S 1

of rby

S is theidealof allthepolynomialsp2R suh thatthere exists apowerprodut hof

elementsofS suhthathp2r.

1.1.

Gr

obnerbases

InthissetionR=K[X℄denotesapolynomialringoveraeld.Weonlyreallsome

propertiesofGrobnerbases.ThebooksofCoxetal.(1992)andBekerandWeispfenning

(1991)providearealpresentation.

IfB isaGrobnerbasisofanidealrofapolynomialringR=K[X℄foranorderingR.

TheredutionbyB,denoted

!

B

preservestheequivalenerelationmodrand wehave

1 r=(B),

2 whenitisredued,aGrobnerbasisisaanonialrepresentativeofr(itonlydepends

ontheidealandontheordering),

3 the ideal r is equal to R if and only if 1 2 B (Beker and Weispfenning, 1991,

orollary6.16,page257),

4 given any p2 R ,there exists aunique polynomial pirreduible by B suh that

p

!

B

p. This polynomial is a anonial representative of the residue lass of p

modulo r (it only depends on the idealand theordering). Inpartiular, ifp 2 r

thenp=0,

Even ifX is innite, one anompute Grobner bases of nitely generated ideals of

K[X℄.This remark is importantsine we aregoing to omputeGrobnerbases of (non

dierential) idealsin dierentialpolynomialrings. Thetheoretial justiationis given

bythefollowinglemma.

Lemma 1.1. Letrbe anideal of aring Randx betransendentaloverR .If denotes

the anonial ring homomorphism :R!R [x℄then 1

(r)=r.

LetA = fp

1

; :::; p

n

g and S =fs

1

;:::;s

m

gbenite sets of polynomialsof R . Let

fz

1

;:::;z

m

g be anite set of indeterminates overR . Onegets aGrobnerbasis B

0 of

theidealS 1

(A)ofS 1

R byomputingaGrobnerbasisoftheset

fp

1

;:::;p

n

;s

1 z

1

1;:::;s

m z

m 1g

foranyordering(Eisenbud,1995,exerise2.2,page79).Eahz

i

standsfor1=s

i

.Toget

aGrobnerbasisB

1 of(A)

:

S 1

,omputerstB

0

foranyordering whiheliminatesthe

z's.ThenB

1

=B

0

\R (BekerandWeispfenning,1991,proposition 6.15,page257).

2. Lazard'slemma

Lazard'slemma(theorem2.1)isaresultofommutativealgebra,interestinginitself.