New order bounds in differential elimination algorithms

0.18 MB·

by  Richard Gustavson

✩
New order bounds in differential elimination algorithms
RichardGustavson
CUNYGraduateCenter,Ph.D.PrograminMathematics,365FifthAvenue,NewYork,NY10016,USA
AlexeyOvchinnikov
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1 CUNYQueensCollege,DepartmentofMathematics,65-30KissenaBoulevard,Queens,NY11367,USA
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2 GlebPogudin

v JohannesKeplerUniversity,InstituteforAlgebra,ScienceParkII,3rdFloor,4040Linz,Austria
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Abstract
]
C
WepresentanewupperboundfortheordersofderivativesintheRosenfeld-Gro¨bneralgorithm.
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This algorithm computes a regular decomposition of a radical differential ideal in the ring of
.
h differentialpolynomialsoveradifferentialfield ofcharacteristiczerowithan arbitrarynumber
t of commutingderivations. Thisdecompositioncan then be used to test formembershipin the
a
m given radical differential ideal. In particular, this algorithm allows us to determine whether a
systemofpolynomialPDEsisconsistent.
[
Previously, the only known order upper bound was given by Golubitsky, Kondratieva,

2 MorenoMaza, and Ovchinnikovfor the case of a single derivation. We achieve our boundby
v
associatingtothealgorithmantichainsequenceswhoselengthscanbeboundedusingtheresults
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ofLeo´nSa´nchezandOvchinnikov.
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2 Keywords: Polynomialdifferentialequations;differentialeliminationalgorithms;
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computationalcomplexity
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.
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1 1. Introduction
:
v
TheRosenfeld-Gro¨bneralgorithmisafundamentalalgorithminthealgebraictheoryofdif-
i
X ferentialequations. Thisalgorithm,whichfirstappearedin(Boulieretal.,1995,2009),takesas
r itsinputafinitesetF ofdifferentialpolynomialsandoutputsarepresentationoftheradicaldif-
a ferentialidealgeneratedbyF asafiniteintersectionofregulardifferentialideals. Thealgorithm
has many applications; for example, it can be used to test membershipin a radicaldifferential
ideal,and,inconjunctionwiththedifferentialNullstellensatz,canbeusedtotesttheconsistency
✩ Thisworkwaspartially supported bytheNSFgrants CCF-095259, CCF-1563942, DMS-1606334, bythe NSA
grant#H98230-15-1-0245,byCUNYCIRG#2248,byPSC-CUNYgrant#69827-0047,bytheAustrianScienceFund
FWFgrantY464-N18.
Emailaddresses:[email protected](RichardGustavson),[email protected]
(AlexeyOvchinnikov),[email protected](GlebPogudin)
PreprintsubmittedtoElsevier December1,2016
of a system of polynomialdifferentialequations. See (Golubitskyetal., 2008) for a history of
thedevelopmentoftheRosenfeld-Gro¨bneralgorithmandsimilardecompositionalgorithms.
The Rosenfeld-Gro¨bner algorithm has been implemented in Maple as a part of the
DifferentialAlgebrapackage. In order to determine the complexity of the algorithm, we
need to (among other things) find an upper bound on the orders of derivatives that appear in
allintermediatestepsandin theoutputofthealgorithm. Thefirststep inansweringthisques-
tionwascompletedin(Golubitskyetal.,2008),inwhichanupperboundinthecaseofasingle
derivationandanyrankingonthesetofderivativeswasfound.Iftherearenunknownfunctions
andtheorderoftheoriginalsystemish,theauthorsshowedthatanupperboundontheorders
oftheoutputoftheRosenfeld-Gro¨bneralgorithmish(n 1)!.
−
In this paper, we extend this result by finding an upperboundfor the ordersof derivatives
thatappearintheintermediatestepsandintheoutputoftheRosenfeld-Gro¨bneralgorithminthe
caseofanarbitrarynumberofcommutingderivationsandaweightedrankingonthederivatives.
We first compute an upper bound for the weights of the derivatives involved for an arbitrary
weightedranking;bychoosingaspecificweight,weobtainanupperboundfortheordersofthe
derivatives.Forthis,weconstructspecialantichainsequencesinthesetZm 1,...,n equipped
>0×{ }
withaspecificpartialorder.Wethenuse(Leo´nSa´nchezandOvchinnikov,2016)toestimatethe
lengthsofoursequences. Ageneralanalysisoflengthsofantichainsequencesbeganin(Pierce,
2014)andcontinuedin(FreitagandLeo´nSa´nchez,2016).
Weshowthatanupperboundfortheweightsofderivativesintheintermediatestepsandin
theoutputoftheRosenfeld-Gro¨bneralgorithmisgivenbyhf ,wherehistheweightofourin-
L+1
putsystemofdifferentialequations, f , f , f ,... istheFibonaccisequence 0,1,1,2,3,5,... ,
0 1 2
{ } { }
and Listhemaximalpossiblelengthofacertainantichainsequence(thatdependssolelyonh,
thenumbermofderivations,andthenumbernofunknownfunctions).Form=2,werefinethis
upperboundinanewwaybyshowingthattheweightsofthederivativesinquestionarebounded
abovebyasequencedefinedsimilarlytotheFibonaccisequencebutwithaslowergrowthrate.
Bychoosingaspecificweight,weareabletoproduceanupperboundfortheordersofthe
derivativesintheintermediatestepsandintheoutputoftheRosenfeld-Gro¨bneralgorithm.Note
that this bound is different from the upper bounds for the effective differential Nullstellensatz
(D’Alfonsoetal.,2014;Gustavsonetal.,2016a),whicharehigherandalsodependonthedegree
ofthegivensystemofdifferentialequations. Ourresultisanimprovementof(Gustavsonetal.,
2016b) because it allows us to compute sharper order upper bounds with respect to specific
derivationsthanthepreviousupperbounddid,andbecauseoftherefinementinthecasem = 2.
Forexample,ifn=2andh=3,4,5,thenewboundis3,8,33timesbetter,respectively.
Thepaperisorganizedasfollows.InSection2,wepresentthebackgroundmaterialfromdif-
ferentialalgebrathatisnecessarytounderstandtheRosenfeld-Gro¨bneralgorithm. InSection3,
wedescribethisalgorithmasitispresentedin(Hubert,2003),aswellastwonecessaryauxiliary
algorithms.InSection4,weproveourmainresultontheupperbound.InSection5,wecalculate
theupperboundforspecificvaluesusingtheresultsof(Leo´nSa´nchezandOvchinnikov,2016).
InSection6, wegiveanexampleshowingthatthelowerboundfortheordersofderivativesin
theRosenfeld-Gro¨bneralgorithmisatleastdouble-exponentialinthenumberofderivations.
2. Backgroundondifferentialalgebra
Inthissection,wepresentbackgroundmaterialfromdifferentialalgebrathatispertinentto
theRosenfeld-Gro¨bneralgorithm.Foramorein-depthdiscussion,wereferthereaderto(Hubert,
2003;Kolchin,1973).
2
Definition 1. A differential ring is a commutative ring R with a collection of m commuting
derivations∆= ∂ ,...,∂ onR.
1 m
{ }
Definition2. AnidealI ofadifferentialringisadifferentialidealifδa Iforalla I,δ ∆.
∈ ∈ ∈
ForasetA R,let(A), √(A),[A],and A denotethesmallestideal,radicalideal,differential
⊆ { }
ideal,andradicaldifferentialidealcontainingA,respectively.IfQ R,then A = √[A].
⊆ { }
Remark3. Inthispaper,asusual,wealsousethebraces a ,a ,... todenotethesetcontaining
1 2
{ }
the elementsa ,a ,.... Even thoughthisnotationconflictswith the abovenotationforradical
1 2
differentialideals(usedhereforhistoricalreasons),itwillbeclearfromthecontextwhichofthe
twoobjectswemeanineachparticularsituation.
In this paper, k is a differential field of characteristic zero with m commuting derivations.
Thesetofderivativeoperatorsisdenotedby
Θ:= ∂i11···∂imm :ij ∈Z>0,16 j6m .
n o
ForY = y ,...,y asetofndifferentialindeterminates,thesetofderivativesofY is
1 n
{ }
ΘY := θy:θ Θ,y Y .
{ ∈ ∈ }
Thentheringofdifferentialpolynomialsoverkisdefinedtobe
k Y =k y ,...,y :=k[θy:θy ΘY].
1 n
{ } { } ∈
Wecannaturallyextendthederivations∂ ,...,∂ totheringk Y bydefining
1 m
{ }
∂j ∂i11···∂immyk :=∂i11···∂ijj+1···∂immyk.
(cid:16) (cid:17)
Foranyθ=∂i1 ∂im Θ,wedefinetheorderofθtobe
1 ··· m ∈
ord(θ):=i + +i .
1 m
···
Foranyderivativeu=θy ΘY,wedefine
∈
ord(u):=ord(θ).
Foradifferentialpolynomial f k Y k,wedefinetheorderof f tobethemaximumorderof
∈ { }\
allderivativesthatappearin f. ForanyfinitesetA k Y k,weset
⊆ { }\
(A):=max ord(f): f A . (1)
H { ∈ }
Foranyθ=∂i1 ∂im andpositiveintegersc ,...,c Z ,wedefinetheweightofθtobe
1 ··· m 1 m ∈ >0
w(θ)=w ∂i11···∂imm :=c1i1+···+cmim.
(cid:16) (cid:17)
Note thatif all ofthe c = 1, then w(θ) = ord(θ) forall θ Θ. For a derivativeu = θy ΘY,
i
∈ ∈
wedefinetheweightofutobew(u) := w(θ). Foranydifferentialpolynomial f k Y k,we
∈ { }\
definetheweightof f, w(f),tobethemaximumweightofallderivativesthatappearin f. For
anyfinitesetA k Y k,weset
⊆ { }\
(A):=max w(f): f A .
W { ∈ }
3
Definition4. ArankingonthesetΘY isatotalorder<satisfyingthefollowingtwoadditional
properties: forallu,v ΘY andallθ Θ,θ,id,
∈ ∈
u<θu and u<v = θu<θv.
⇒
Aranking<iscalledanorderlyrankingifforallu,v ΘY,
∈
ord(u)<ord(v) = u<v.
⇒
Givenaweightw,aranking<onΘY iscalledaweightedrankingifforallu,v ΘY,
∈
w(u)<w(v) = u<v.
⇒
Remark 5. Note thatif w ∂i1 ∂im = i + +i (thatis, w(θ) = ord(θ)), then a weighted
1 ··· m 1 ··· m
ranking<onΘY isinfacta(cid:16)norderly(cid:17)ranking.
Fromnowon,wefixaweightedranking<onΘY.
Definition6. Let f k Y k.
∈ { }\
The derivative u ΘY of highestrank appearingin f is called the leader of f, denoted
• ∈
lead(f).
If we write f as a univariate polynomialin lead(f), the leading coefficient is called the
•
initialof f,denotedinit(f).
Ifweapplyanyderivativeδ ∆to f,theleaderofδf isδ(lead(f)),andtheinitialofδf is
• ∈
calledtheseparantof f,denotedsep(f).
GivenasetA k Y k,wewilldenotethesetofleadersofAbyL(A),thesetofinitialsof
⊆ { }\
AbyI ,andthesetofseparantsofAbyS ;wethenletH = I S bethesetofinitialsand
A A A A A
∪
separantsofA.
Foraderivativeu ΘY,welet(ΘY) (respectively,(ΘY) )bethecollectionofallderiva-
<u 6u
∈
tivesv ΘYwithv<u(respectively,v6u).Foranyderivativeu ΘY,weletA (respectively,
<u
∈ ∈
A )betheelementsofAwithleader<u(respectively,6u),thatis,
6u
A := A k[(ΘY) ] and A := A k[(ΘY) ].
<u <u 6u 6u
∩ ∩
Wecansimilarlydefine(ΘA) and(ΘA) ,where
<u 6u
ΘA:= θf :θ Θ, f A .
{ ∈ ∈ }
Given f k Y ksuchthatdeg (f)=d,wedefinetherankof f tobe
∈ { }\ lead(f)
rank(f):=lead(f)d.
The weighted ranking < on ΘY determines a pre-order(that is, a relation satisfying all of the
propertiesofanorder,exceptforthepropertythata6bandb6aimplythata=b)onk Y k:
{ }\
Definition7. Given f , f k Y k,wesaythat
1 2
∈ { }\
rank(f )<rank(f )
1 2
iflead(f )<lead(f )oriflead(f )=lead(f )anddeg (f )<deg (f ).
1 2 1 2 lead(f1) 1 lead(f2) 2
4
Definition8. Adifferentialpolynomial f ispartiallyreducedwithrespecttoanotherdifferential
polynomialgifnoproperderivativeoflead(g)appearsin f,and f isreducedwithrespecttog
if,inaddition,
deg (f)<deg (g).
lead(g) lead(g)
A differential polynomialis then (partially) reduced with respect to a set A k Y k if it is
⊆ { }\
(partially)reducedwithrespecttoeveryelementofA.
Definition9. ForasetA k Y k,wesaythatAis:
⊆ { }\
autoreducedifeveryelementofAisreducedwithrespecttoeveryotherelement.
•
weakd-triangularifL(A)isautoreduced.
•
d-triangular if A is weak d-triangular and every element of A is partially reduced with
•
respecttoeveryotherelement.
Notethateveryautoreducedsetisd-triangular. Everyweakd-triangularset(andthusevery
d-triangularandautoreducedset)isfinite(Hubert,2003,Proposition3.9).Sincethesetofleaders
ofaweakd-triangularsetAisautoreduced,distinctelementsofAmusthavedistinctleaders. If
u ΘY istheleaderofsomeelementofaweakd-triangularsetA,weletA denotethiselement.
u
∈
Definition 10. We define a pre-order on the collection of all weak d-triangular sets, which
we also call rank, as follows. Given two weak d-triangular sets A = A ,...,A and B =
1 r
{ }
B ,...,B ,ineachcasearrangedinincreasingrank,wesaythatrank(A)<rank(B)ifeither:
1 s
{ }
thereexistsak 6min(r,s)suchthatrank(A)=rank(B)forall16i<kandrank(A )<
i i k
•
rank(B ),or
k
r> sandrank(A)=rank(B)forall16i6 s.
i i
•
Wealsosaythatrank(A)=rank(B)ifr= sandrank(A)=rank(B)forall16i6r.
i i
We can restrict this rankingto the collection of all d-triangularsets or the collection of all
autoreducedsets.
Definition11. AcharacteristicsetofadifferentialidealIisanautoreducedsetC Iofminimal
⊆
rankamongallautoreducedsubsetsofI.
GivenafinitesetS k Y , letS denotethemultiplicativesetcontaining1andgenerated
∞
⊆ { }
byS. ForanidealI k Y ,wedefinethecolonidealtobe
⊆ { }
I :S := a k Y : s S withsa I .
∞ ∞
{ ∈ { } ∃ ∈ ∈ }
IfI isadifferentialideal,thenI :S isalsoadifferentialideal(Kolchin,1973,SectionI.2).
∞
Definition 12. For a differential polynomial f k Y and a weak d-triangularset A k Y ,
∈ { } ⊆ { }
a differential partial remainder f and a differential remainder f of f with respect to A are
1 2
differentialpolynomialssuchthatthereexist s S , h H suchthat sf f mod [A]and
∈ ∞A ∈ A∞ ≡ 1
hf f mod [A],with f partiallyreducedwithrespecttoAand f reducedwithrespecttoA.
2 1 2
≡
5
We denotea differentialpartialremainderof f with respectto A bypd-red(f,A)anda dif-
ferential remainder of f with respect to A by d-red(f,A). There are algorithms to compute
pd-red(f,A)andd-red(f,A)forany f and A (Hubert, 2003, Algorithms3.12and3.13). These
algorithmshavethepropertythat
rank(pd-red(f,A)), rank(d-red(f,A))6rank(f);
sincewehaveaweightedranking,thisimpliesthat
w(pd-red(f,A)), w(d-red(f,A))6w(f).
Definition 13. Two derivatives u,v ΘY are said to have a common derivative if there exist
∈
φ,ψ Θsuchthatφu = ψv. Notethisisthecasepreciselywhenu = θ yandv = θ yforsome
1 2
∈
y Y andθ ,θ Θ.
1 2
∈ ∈
Definition14. Ifu=∂i1 ∂imyandv=∂j1 ∂jmyforsomey Y,wedefinetheleastcommon
1 ··· m 1 ··· m ∈
derivativeofuandv,denotedlcd(u,v),tobe
lcd(u,v)=∂max(i1,j1) ∂max(im,jm)y.
1 ··· m
Definition15. For f,g k Y k, we definethe∆-polynomialof f andg, denoted∆(f,g), as
∈ { }\
follows. If lead(f) and lead(g) have no common derivatives, set ∆(f,g) = 0. Otherwise, let
φ,ψ Θbesuchthat
∈
lcd(lead(f),lead(g))=φ(lead(f))=ψ(lead(g)),
anddefine
∆(f,g):=sep(g)φ(f) sep(f)ψ(g).
−
Definition16. Apair(A,H)iscalledaregulardifferentialsystemif:
Aisad-triangularset
•
HisasetofdifferentialpolynomialsthatareallpartiallyreducedwithrespecttoA
•
S H
A ∞
• ⊆
forall f,g A,∆(f,g) ((ΘA) ):H ,whereu=lcd(lead(f),lead(g)).
<u ∞
• ∈ ∈
Definition17. Anyidealoftheform[A] : H ,where(A,H)isaregulardifferentialsystem, is
∞
calledaregulardifferentialideal.
Everyregulardifferentialidealisaradicaldifferentialideal(Hubert,2003,Theorem4.12).
Definition18. GivenaradicaldifferentialidealI k Y ,aregulardecompositionofIisafinite
⊆ { }
collectionofregulardifferentialsystems (A ,H ),...,(A ,H ) suchthat
1 1 r r
{ }
r
I = [A]: H .
i i∞
\i=1
DuetotheRosenfeld-Gro¨bneralgorithm,everyradicaldifferentialidealink Y hasaregular
{ }
decomposition.
6
Definition19. Ad-triangularsetC iscalledadifferentialregularchainifitisacharacteristic
setof[C]:H ;inthiscase,wecall[C]:H acharacterizabledifferentialideal.
C∞ C∞
Definition20. AcharacteristicdecompositionofaradicaldifferentialidealI k Y isarepre-
⊆ { }
sentationofI asanintersectionofcharacterizabledifferentialideals.
AswewillrecallinSection3,everyradicaldifferentialidealalsohasacharacteristicdecom-
position.
3. Rosenfeld-Gro¨bneralgorithm
BelowwereproducetheRosenfeld-Gro¨bneralgorithmfrom(Hubert,2003,Section6).This
algorithmrelieson two others, called auto-partial-reduce and update, whichwe also include.
Weincludethesetwoauxiliaryalgorithmsbecause,inSection4,wewillstudytheireffectonthe
growthoftheweightsofderivativesinRosenfeld-Gro¨bner.
Rosenfeld-Gro¨bnertakesasitsinputtwofinitesubsetsF,K k Y andoutputsafiniteset
∈ { }
ofregulardifferentialsystemssuchthat
A
F :K = [A]:H , (2)
∞ ∞
{ }
(A,\H)
∈A
where =∅if1 F :K .
∞
A ∈{ }
If we have a decomposition of F : K as in (2), we can compute, using only algebraic
∞
{ }
operations,adecompositionoftheform
F :K = [C]:H , (3)
{ } ∞ C∞
C\
∈C
where is finite and eachC is a differentialregular chain (Hubert, 2003, Algorithms7.1
C ∈ C
and7.2).Thismeansthatanupperboundon (A H)from(2)willalsobeanupper
bouRndosoennSfeCld∈C-GWro¨(bCn)efrrohmas(3m)a.ny immediatSe (aAp,Hp)l∈iAcaWtions.∪For example, if K = 1 , then
{ }
F : K = F , sointhiscase,Rosenfeld-Gro¨bnercomputesaregulardecompositionof F ,
∞
{ } { } { }
whichthenalsogivesusacharacteristicdecompositionof F bythediscussionintheprevious
{ }
paragraph.
TheweakdifferentialNullstellensatzsaysthatasystemofpolynomialdifferentialequations
F =0isconsistent(thatis,hasasolutioninsomedifferentialfieldextensionofk)ifandonlyif
1<[F](Kolchin,1973,SectionIV.2). Thus,sinceRosenfeld-Gro¨bner(F,K)=∅ifandonlyif
1 F :K ,weseethatF =0isconsistentifandonlyifRosenfeld-Gro¨bner(F, 1 ),∅.
∞
∈{ } { }
Moregenerally,Rosenfeld-Gro¨bneranditsextensionforcomputingacharacteristicdecom-
position of a radical differential ideal allow us to test for membership in a radical differential
ideal,asfollows. Supposewehavecomputedacharacteristicdecomposition
F = [C]:H .
{ } C∞
C\
∈C
Now,adifferentialpolynomial f k Y iscontainedin F ifandonlyif f [C] : H forall
∈ { } { } ∈ C∞
C ;thislattercaseistrueifandonlyifd-red(f,C) = 0,whichcanbetestedusing(Hubert,
∈ C
2003,Algorithm3.13).
Rosenfeld-Gro¨bner,auto-partial-reduce,andupdaterelyonthefollowingtuplesofdiffer-
entialpolynomials:
7
Definition 21. A Rosenfeld-Gro¨bnerquadruple(or RG-quadruple)is a 4-tuple (G,D,A,H)of
finitesubsetsofk Y suchthat:
{ }
Aisaweakd-triangularset,H H,Disasetof∆-polynomials,and
A
• ⊆
forall f,g A,either∆(f,g)=0or∆(f,g) Dor
• ∈ ∈
∆(f,g) (Θ(A G) ):H ,
∈ ∪ <u u∞
whereu=lcd(lead(f),lead(g))andH = H (H H ) k[(ΘY) ].
u A<u ∪ \ A ∩ <u
Algorithm:Rosenfeld-Gro¨bner,(Hubert,2003,Algorithm6.11)
Data: F,K finitesubsetsofk Y
{ }
Result: Aset ofregulardifferentialsystemssuchthat:
A
isemptyifithasbeendetectedthat1 F : K
∞
• A ∈{ }
F : K = [A]:H otherwise
∞ ∞
• { }
(A,H)
T∈A
:= (F,∅,∅,K) ;
S { }
:=∅;
A
while ,∅do
S
(G,D,A,H):=anelementof ;
S
¯ = (G,D,A,H);
S S\
if G D=∅then
∪
:= auto-partial-reduce(A,H);
A A∪
else
p:=anelementofG D;
G¯,D¯ :=G p ,D p∪;
\{ } \{ }
p¯ :=d-red(p,A);
if p¯ =0then
¯ := ¯ (G¯,D¯,A,H) ;
S S∪{ }
else
if p¯ <kthen
p¯ := p¯ init(p¯)rank(p¯) p¯ :=deg (p¯)p¯ lead(p¯)sep(p¯);
i − s lead(p¯) −
¯ := ¯ update(G¯,D¯,A,H,p¯),(G p¯ ,sep(p¯) ,D¯,A,H
s
Sinit(pS¯)∪),{(G¯ p¯ ,init(p¯) ,D¯,A,H)∪; { } ∪
i
{ } ∪{ } }
end
end
end
:= ¯;
S S
end
return ;
A
Remark22. TheRG-quadruplethatisoutputbyupdatesatisfiesadditionalpropertiesthatwe
donotlist,astheyarenotimportantforouranalysis. Formoreinformation,wereferthereader
to(Hubert,2003,Algorithm6.10)
8
Algorithm:auto-partial-reduce,(Hubert,2003,Algorithm6.8)
Data:TwofinitesubsetsA,Hofk Y suchthat(∅,∅,A,H)isanRG-quadruple
{ }
Result:
Theemptysetifitisdetectedthat1 [A]:H
∞
• ∈
Otherwise,asetwithasingleregulardifferentialsystem(B,K)withL(A)=L(B),
•
H K,and[A]:H =[B]: K
B ∞ ∞
⊆
B:=∅;
for u L(A)increasinglydo
∈
b:=pd-red(A ,B);
u
if rank(b)=rank(A )then
u
B:= B b ;
∪{ }
else
return(∅);
end
end
K := H pd-red(p,B): p H H ;
B A
∪{ ∈ \ }
if 0 K then
∈
return(∅);
else
return (B,K) ;
{ }
end
Algorithm:update(Hubert,2003,Algorithm6.10)
Data:
A4-tuple(G,D,A,H)offinitesubsetsofk Y
• { }
AdifferentialpolynomialpreducedwithrespecttoAsuchthat(G p ,D,A,H)isan
• ∪{ }
RG-quadruple
Result: AnewRG-quadruple(G¯,D¯,A¯,H¯)
u:=lead(p);
G := a A lead(a) Θu ;
A
A¯ := A{ G∈ ; | ∈ }
A
G¯ :=G\ G ;
A
D¯ := D∪ ∆(p,a) a A¯ 0 ;
H¯ := H∪{sep(p),|init∈(p)};\{ }
return (∪G¯{,D¯,A¯ p ,H¯)};
∪{ }
9
4. Orderupperbound
GivenfinitesubsetsF,K k Y ,leth= (F K). Ourgoalistofindanupperboundfor
⊆ { } W ∪
(A H) ,
W ∪ 
(A,[H)∈A 
where = Rosenfeld-Gro¨bner(F,K),intermsofh,m(thenumberofderivations),andn(the
A
numberofdifferentialindeterminates).Bythenchoosingaspecificweight,wecanfindanupper
boundfor (A H) intermsofm,n,and (F K).
WeappHro(cid:16)aSch(At,hHi)s∈Aprob∪lema(cid:17)sfollows. Every(A,HH) ∪isformedbyapplyingauto-partial-
∈A
reducetoa4-tuple(∅,∅,A,H ) . Thus,itsuffices:
′ ′
∈S
toboundhowauto-partial-reduceincreasestheweightofacollectionofdifferentialpoly-
•
nomials(itturnsouttonotincreasetheweight),and
to bound (G D A H) for all (G,D,A,H) added to throughout the course of
• W ∪ ∪ ∪ S
Rosenfeld-Gro¨bner.
We accomplishthelatterbydeterminingwhentheweightofa tuple(G,D,A,H)addedto is
S
largerthantheweightsofthepreviouselementsof andbounding (G D A H)inthis
S W ∪ ∪ ∪
instance,andthenboundingthenumberoftimeswecanaddsuchelementsto .
S
There is a sequence (G,D,A,H) N corresponding to each regular differential system
{ i i i i }i=0
(A,H)intheoutputofRosenfeld-Gro¨bner,whereN = N ,suchthat(G ,D ,A ,H )
(A,H) i+1 i+1 i+1 i+1
is obtained from (G,D,A,H) during the while loop, (G ,D ,A ,H ) = (F,∅,∅,K), and
i i i i 0 0 0 0
(A,H)=auto-partial-reduce(A ,H ).
N N
We begin with an auxiliary result, which is an analogue of the first property from
(Golubitskyetal.,2009,Section5.1).
Lemma 23. Forevery f A andi < j, there exists g A such thatlead(f) Θlead(g). In
i j
∈ ∈ ∈
particular,if pisreducedwithrespecttoA ,then pisreducedwithrespecttoA foralli< j.
j i
Proof. Itissufficienttoconsiderthecase j=i+1. If(G ,D ,A ,H )wasobtainedfrom
i+1 i+1 i+1 i+1
(G,D,A,H)withoutapplyingupdate,thenA = A . Otherwise,either f A G (weuse
i i i i i i+1 ∈ i\ Ai
thenotationfromupdate),or f G . Intheformercase, f A aswell,sowecansetg= f.
∈ Ai ∈ i+1
Inthelattercase,lead(f) Θlead(p),sowecansetg= p.
∈
We definea partialorder4onthe setofderivativesΘY asfollows. Foru,v ΘY, we say
∈
that u 4 v if there exists θ Θ such that θu = v. Note that this implies thatu and v are both
∈
derivativesofthesamey Y.
∈
Definition 24. An antichainsequencein ΘY is a sequenceofelements S = s ,s ,... ΘY
1 2
{ } ⊆
thatarepairwiseincomparableinthispartialorder.
Givenasequence (G,D,A,H) N asabove(whereN = N forsomeregulardifferential
{ i i i i }i=0 (A,H)
system (A,H) in the output of Rosenfeld-Gro¨bner), we will construct an antichain sequence
S = s ,s ,... ΘY inductivelygoingalongthe sequence (G,D,A,H) . SupposeS =
1 2 i i i i j 1
{ } ⊆ { } −
s ,...,s hasbeenconstructedafterconsidering(G ,D ,A ,H ),...,(G ,D ,A ,H ),
1 j 1 0 0 0 0 i 1 i 1 i 1 i 1
w{ hereS −=}∅. A4-tuple(G,D,A,H)canbeobtainedfromthetuple(G− ,D− ,A− ,H− )
0 i i i i i 1 i 1 i 1 i 1
− − − −
intwoways:
10