Table Of ContentComputing the differential Galois group of a parameterized
second-order linear differential equation
Carlos E. Arreche
ABSTRACT
4 We develop algorithms to compute the differential Galois group G associated to a parameterized
1
second-order homogeneous lineardifferential equationoftheform
0
2 ¶ 2 ¶
Y+r Y +r Y =0,
¶ x2 1¶ x 0
n
a wherethecoefficientsr1,r0∈F(x)arerationalfunctionsinxwithcoefficientsinapartialdifferential
J
fieldF ofcharacteristic zero. Ourworkrelies ontheprocedure developed by Dreyfustocompute G
0
undertheassumption thatr =0.Weshowhowtocompletethisprocedure tocoverthecaseswhere
2 1
r16=0,byreinterpreting aclassical changeofvariablesprocedure inGalois-theoretic terms.
]
C
1. Introduction
A
Consideralineardifferential equation oftheform
.
h
t n−1
a d nY +(cid:229) rd iY =0, (1.1)
m x i x
i=0
[
wherer ∈K:=F(x),thefieldofrationalfunctionsinxwithcoefficientsinaP -fieldF,d denotesthederivative
i x
1 with respect to x, and P :={¶ ,...,¶ } is a set of commuting derivations. Letting D :={d }∪P , consider K
1 m x
v
as a D -field by setting ¶ x=0 for each j. The parameterized Picard-Vessiot theory of [5] associates a parame-
7 j
2 terizedPicard-Vessiot(PPV)grouptosuchanequation.InanalogywiththePicard-Vessiottheorydevelopedby
1 Kolchin[12],thePPV-groupmeasurestheP -algebraic relationsamongstthesolutions to(1.1).Thedifferential
5
. Galoisgroupsthatariseinthistheoryarelineardifferential algebraicgroups:subgroups ofGLn thataredefined
1
by the vanishing of systems of polynomial differential equations in the matrix entries. The study of linear dif-
0
4 ferential algebraicgroups waspioneered in[4].Theparameterized Picard-Vessiot theoryof[5]isaspecial case
1 of an earlier generalization of Kolchin’s theory, presented in [17], as well as the differential Galois theory for
:
v difference-differential equations withparameters [10].
i
X Thisworkaddresses theexplicitcomputation ofthePPV-groupGcorresponding toasecond-order parame-
r terizedlineardifferential equation
a
d 2Y+r d Y+r Y =0, (1.2)
x 1 x 0
where r ,r ∈F(x)=:K, and F is a P -field. In [7], Dreyfus applies results from [6] to develop algorithms to
1 0
compute G, under the assumption that r =0 (see also [1] for a detailed discussion of Dreyfus’ results in the
1
setting of one parametric derivation, and [3] for the computation of the unipotent radical). We complete these
algorithms to compute G when r is not necessarily zero. Algorithms for higher-order equations are developed
1
in[18,19].
After performing a change of variables on (1.2), we obtain an associated equation (3.1) of the form d 2Y −
x
qY =0,whosePPV-groupH isalreadyknown[3,7].In§3,wereinterpretthischange-of-variables procedurein
termsofalattice (3.6)ofPPV-fields. Werecovertheoriginal PPV-group Gfrom thislattice inProposition 3.3,
whichisaformalconsequence oftheparameterized Galoiscorrespondence [5,Thm.3.5].Thisreinterpretation,
whose non-parameterized analogue is probably well-known to the experts in classical Picard-Vessiot theory,
comprisesatheoretical procedure torecoverthePPV-groupof(1.2)fromthesedata.
2010MathematicsSubjectClassificationPrimary34M15;Secondary12H20,34M03,20H20,13N10,37K20
Keywords:Parameterizeddifferentialequation,parameterizedPicard-Vessiottheory,lineardifferentialalgebraicgroup.
ThismaterialisbaseduponworkpartiallysupportedbyaNationalScienceFoundation(NSF)GraduateResearchFellowship(grant
40017-04-05)andbyNSFgrantCCF-0952591.
CARLOS E. ARRECHE
In §4, the main tools leading to the explicit computation of G obtained in Theorem 4.1 are Proposition 3.3
and the Kolchin-Ostrowski Theorem [13]. This strategy for computing G was already sketched in [1, §3.4],
in the setting of one parametric derivation. The results here are sharper than those of [1], and the proofs are
conceptually simpler.
In§5,weapplyTheorem4.1andtheresultsof[3,7]tocomputethePPV-groupcorresponding toaconcrete
parameterized second-order lineardifferential equation (5.1).
2. Preliminaries
Wereferto[14,20]formoredetails concerning thefollowingdefinitions. Everyfieldconsidered inthisworkis
assumedtobeofcharacteristiczero.AringRequippedwithafinitesetD :={d ,...,d }ofpairwisecommuting
1 m
derivations (i.e.,d (ab)=ad (b)+d (a)bandd d =d d foreach a,b∈K and16i,j6m)iscalled aD -ring.
i i i i j j i
IfR=K happens tobe afield, wesay that (K,D )isaD -field. Weoften omit the parentheses, and simply write
d a for d (a). For P ⊆D , we denote the subring of P -constants of R by RP :={a∈K | d a=0, d ∈P }. When
P ={d }isasingleton, wewriteRd insteadofRP .
Theringofdifferential polynomials overK (inmdifferential indeterminates) isdenoted byK{Y1,...,Ym}D .
Asaring,itisthefreeK-algebrainthecountably infinitesetofvariables
{q Y |16i6m, q ∈Q }; where
i
Q :={d r11...d rnn |ri∈Z>0 for 16i6n}
isthefreecommutativemonoidonthesetD .TheringK{Y1,...,Ym}D carriesanaturalstructureofD -ring,given
by d i(q Yj):= (d i·q )Yj. We say p∈ K{Y1,...,Ym}D is a linear differential polynomial if it belongs to the K-
vectorspacespannedbytheq Y ,forq ∈Q and16 j6m.TheK-vectorspaceoflineardifferentialpolynomials
j
will be denoted by K{Y ,...,Y }1. The ring of linear differential operators K[D ] is the K-span of Q , and its
1 m D
(non-commutative) ringstructure isdefinedbycomposition ofadditiveendomorphisms ofK.
IfMisaD -fieldandKisasubfieldsuchthatd (K)⊂Kforeachd ∈D ,wesayKisaD -subfieldofMandMis
aD -fieldextensionofK.Ify ,...,y ∈M,wedenotetheD -subfieldofM generatedoverK byallthederivatives
1 n
oftheyi byKhy1,...,yniD .
WesaythataD -fieldK isD -closed ifeverysystem ofpolynomial differential equations defined overK that
admits a solution in some D -field extension of K already has a solution in K. This last notion is discussed at
lengthin[11].Seealso[5,23].
Wenowbrieflyrecallthemainfactsthatwewillneedfromtheparameterized Picard-Vessiot theory[5]and
the theory of linear differential algebraic groups [4,15]. Let F be a P -field, where P :={¶ ,...,¶ }, and let
1 m
K:=F(x)bethefieldofrationalfunctionsinxwithcoefficientsinF,equippedwiththestructureof({d }∪P )-
x
fielddetermined bysetting d xx=1, Kd x =F,and¶ ix=0foreach i.Wewillsometimes refer tod x asthemain
derivation, andtoP asthesetofparametricderivations. Fromnowon,wewillletD :={d }∪P .Consider the
x
followinglineardifferential equation withrespecttothemainderivation, wherer ∈K foreach06i6n−1:
i
n−1
d nY +(cid:229) rd iY =0. (2.1)
x i x
i=0
DEFINITION 2.1. WesaythataD -fieldextension M ⊇K isaparameterized Picard-Vessiot extension (orPPV-
extension) ofK for(2.1)if:
(i) There exist n distinct, F-linearly independent elements y ,...,y ∈M such that d ny +(cid:229) rd iy =0 for
1 n x j i i x j
each16 j6n.
(ii) M=Khy1,...yniD .
d d
(iii) M x =K x.
2
COMPUTING PPVGROUPS OF SECOND-ORDER EQUATIONS
Theparameterized Picard-Vessiotgroup(orPPV-group)isthegroupofD -automorphismsofM overK,and
wedenoteitbyGalD (M/K).TheF-linearspanofalltheyj isthesolution spaceS.
If F is P -closed, it is shown in [5] that a PPV-extension of K for (2.1) exists and is unique up to K-D -
isomorphism. Although this assumption allows for a simpler exposition of the theory, several authors [8,25]
have shown that, in many cases of practical interest, the parameterized Picard-Vessiot theory can be developed
without assuming that F is P -closed. In any case, we may always embed F in a P -closed field [11,23]. The
actionofGalD (M/K)isdeterminedbyitsrestrictiontoS,whichdefinesanembeddingGalD (M/K)֒→GLn(F)
after choosing an F-basis for S. It is shown in [5] that this embedding identifies the PPV-group with a linear
differential algebraic group(Definition2.2),andfromnowonwewillmakethisidentification implicitly.
DEFINITION2.2. LetF beaP -closedfield.WesaythatasubgroupG⊆GLn(F)isalineardifferentialalgebraic
groupifGisdefinedasasubsetofGL (F)bythevanishingofasystemofpolynomial differential equationsin
n
thematrixentries, withcoefficients inF.WesaythatGisP -constant ifG⊆GL (FP ).
n
Thereisaparameterized Galoiscorrespondence [5,Thm.3.5]between thelinear differential algebraic sub-
groups G of GalD (M/K) and the intermediate D -fields K ⊆ L ⊆ M, given by G 7→ MG and L 7→ GalD (M/L).
Underthiscorrespondence, anintermediateD -fieldLisaPPV-extensionofK (forsomelineardifferentialequa-
tion with respect to d x) if and only if GalD (M/L) is normal in GalD (M/K). The restriction homomorphism
GalD (M/K)→GalD (L/K)definedbys 7→s |L issurjective, withkernelGalD (M/L).
ThedifferentialalgebraicsubgroupsoftheadditiveandmultiplicativegroupsofF,whichwedenoterespec-
tivelybyG (F)andG (F),wereclassifiedbyCassidyin[4,Prop.11,Prop.31anditsCorollary]:
a m
PROPOSITION 2.3 (Cassidy). IfB6Ga(F)isadifferentialalgebraicsubgroup,thenthereexistfinitelymany
lineardifferentialpolynomialsp ,...,p ∈F{Y}1 suchthat
1 s P
B={b∈G (F)|p(b)=0foreach16i6s}.
a i
IfA6G (F)isadifferentialalgebraicsubgroup,theneitherA=µ ,thegroupofℓthrootsofunity,orelse
m ℓ
Gm(FP )⊆A,andthereexistfinitelymanylineardifferentialpolynomialsq1,...,qs∈F{Y1,...,Ym}1P suchthat
A= a∈G (F) q ¶ 1a,...,¶ ma =0for16i6s .
m i a a
n (cid:12) o
(cid:12) (cid:0) (cid:1)
(cid:12)
3. Recoveringtheoriginalgroup
Recall that K :=F(x) is the D -field defined by: F =Kd x is P -closed field, D :={d x}∪P , d xx=1, and ¶ x=0
foreach¶ ∈P .Considerasecond-order parameterized lineardifferential equation
d 2Y−qY =0, (3.1)
x
where q∈K. In [7], Dreyfus develops the following procedure to compute the PPV-group H corresponding to
(3.1)(seealso[1,3]).AsinKovacic’salgorithm [16],onefirstdecideswhetherthereexistsu∈K¯ suchthat
(d +u)◦(d −u)=d 2−q, (3.2)
x x x
whereK¯ isanalgebraicclosureofK.Expandingtheleft-handsideof(3.2)showsthatsuchafactorizationexists
precisely when one can find a solution in K¯ to the Riccati equation P (u)=d u+u2−q=0. One can deduce
q x
structuralproperties ofH fromthealgebraicdegreeofsuchauoverK [16].By[7,Thm.2.10],preciselyoneof
thefollowingpossibilities occurs.
I. Ifthere existsu∈K such thatP (u)=0,then thereexistdifferential algebraic subgroups A6G (F)and
q m
B6G (F)suchthatH isconjugate to
a
a b
a∈A, b∈B . (3.3)
(cid:26)(cid:18)0 a−1(cid:19)(cid:12) (cid:27)
(cid:12)
(cid:12)
(cid:12)
3
CARLOS E. ARRECHE
II. If there exists u∈ K¯, of degree 2 over K, such that P (u) = 0, then there exists a differential algebraic
q
subgroup A6G (F)suchthatH isconjugate to
m
a 0 0 −a
a∈A ∪ a∈A .
(cid:26)(cid:18)0 a−1(cid:19)(cid:12) (cid:27) (cid:26)(cid:18)a−1 0 (cid:19)(cid:12) (cid:27)
(cid:12) (cid:12)
(cid:12) (cid:12)
III. Ifthereexistsu∈K¯,ofdegree4,6,or12(cid:12)overK,suchthatP (u)=(cid:12)0,thenH isoneofthefiniteprimitive
q
groupsASL2,SSL2,orASL2,respectively (see[22,§4]).
4 4 5
IV. If there is no u in K¯ such that P (u) =0, then there exists a subset P ′ ⊂F·P consisting of F-linearly
q
independent, pairwisecommutingderivations ¶ ′ suchthatH isconjugate toSL (FP ′).
2
The computation of A in cases I and II is explained in [7]. When A⊆G(FP ) is P -constant, an effective
procedure to compute B in case I is given in [7, §2, p.7]. This procedure is extended to the case when A is
not necessarily P -constant in [3]. The computation of G in case III is reduced to Picard-Vessiot theory [5,
Prop.3.6(2)],andthealgorithmsof[22]canbeusedtocomputeGinthiscase.IncaseIV,forany¶ ∈D:=F·P
it is shown in [7, Prop. 2.8] that H is conjugate to a subgroup of SL (F¶ ) if and only if a certain 3rd-order
2
inhomogeneous linear differential equations with respect to d admits a solution in K, which can be decided
x
effectively (see [20, Prop. 2.24] and [21, §3]). The results of [9] reduce the problem for general P to the one-
parametersituation.
WenowapplytheseresultstocomputethePPV-groupGcorresponding to
d 2Y−2r d Y+r Y =0, (3.4)
x 1 x 0
wherer ,r ∈K,andr isnotnecessarily zero.Theharmlessnormalization −2r insteadofr willspareusthe
1 2 1 1 1
eyesoreofaubiquitous factorof−1 inwhatfollows.
2
Thesolutionsfor(3.4)arerelatedtothesolutionsofanassociatedunimodularequationbyaclassicalchange
ofvariables. Lettingq:=r12−d xr1−r0,letM denoteaPPV-extension ofK for(3.1),soH =GalD (M/K)isthe
corresponding PPV-group,whichweassumehasalready beencomputedby[3,7].
Let{h ,x }denoteabasisforthesolution spaceof(3.1),andletU denote aPPV-extension ofM for
d Y−r Y =0, (3.5)
x 1
andchoose06=z ∈U suchthatd z =r z .Acomputationshowsthat{zh ,zx }isanF-basisforthesolutionspace
x 1
of(3.4),whenceE :=Khzh ,zx iD ⊆U isaPPV-extensionofK for(3.4),anditsPPV-groupisG=GalD (E/K).
Letting N :=Khz iD ⊆U,weseethat N isaPPV-extension ofK for(3.5),and wedenote itsPPV-group by
D(the mnemonic is“determinant”). Thecomputation ofDis analogous tothat ofAincase I(see [7]). Finally,
letR:=M∩N⊆U.SinceD6Gm(F)isabelian,GalD (N/R)6Disnormal,andthereforeRisaPPV-extension
ofK,withPPV-groupdenotedbyL .WeobtainthefollowinglatticeofPPV-extensions andPPV-groups.
U (3.6)
µe ⑦⑦ ❅❅❅
⑦ ❅
⑦⑦⑦ ❅❅❅ where:
⑦
E N M •E isaPPV-extension ofK for(3.4);
❅
❅❅❅❅❅❅❅ ⑦⑦⑦⑦⑦⑦⑦ ••MN iissaaPPPPVV--eexxtteennssiioonnooffKK ffoorr((33..51));;
⑦
R •U isaPPV-extension ofM for(3.5);
G D H
L •R=M∩N isaPPV-extension ofK.
K
4
COMPUTING PPVGROUPS OF SECOND-ORDER EQUATIONS
Remark3.1. Infact,U =Khzh ,zx ,z iD isaPPV-extension ofK for
d 3Y− 3r +d xq d 2Y+ 2r2−2d r +r +2r d xq d Y+ d r −r r −r d xq Y =0. (3.7)
x 1 q x 1 x 1 0 1 q x x 0 1 0 0 q
Toverifythateacho(cid:0)fzh ,zx ,(cid:1)andz sa(cid:0)tisfies(3.7),notethat (cid:1) (cid:0) (cid:1)
d 2z −2r d z +r z =−qz ,
x 1 x 0
andexpandthefollowingproductinK[d ]toobtaintheoperator in(3.7):
x
d −r −d xq ◦(d 2−2r d +r ).
x 1 q x 1 x 0
Let G :=GalD (U/K) be the PPV-grou(cid:0)p ofU over K(cid:1). The choice of D -field generators {h ,x ,z } forU over K
produces thefollowingembeddingofG inGL (F):
3
ag bg 0 g (h )=ag h +cg x ;
g 7→cg dg 0, where g (x )=bg h +dg x ; (3.8)
0 0 eg g (z )=eg z .
EmbeddingGinGL2(F)bymeansofthebasis{zh ,zx },thesurjection G ։Gisthengivenbyg 7→eg · acgg bdgg ,
andtherefore (cid:0) (cid:1)
G≃ eg ag eg bg g ∈G . (3.9)
(cid:26)(cid:18)eg cg eg dg (cid:19)(cid:12) (cid:27)
(cid:12)
(cid:12)
Our next task is to apply the parameterized Galois corr(cid:12)espondence [5, Thm. 3.5] to the lattice (3.6) to
computeG ,andtherefore G,intermsofH andD.Theargumentsarefamiliarfromclassical Galoistheory.
LEMMA 3.2. Therestrictionhomomorphisms
GalD (U/M)→GalD (N/R); and
GalD (U/N)→GalD (M/R),
definedrespectivelybyg 7→g | andg 7→g | ,areisomorphisms.
N M
Proof. SinceU =M·N,these homomorphisms are injective. Since the image of GalD (U/M)in GalD (N/R)is
Kolchin-closed, itsfixedfieldR′ isanintermediate D -fieldextension R⊆R′⊆N.Sinceevery f ∈R′ isfixedby
every g ∈GalD (U/M), it follows that f ∈M, whence f ∈R. By [5, Thm. 3.5], the image of GalD (U/M) must
beallofGalD (N/R).Thesurjectivity ofGalD (U/N)→GalD (M/R)isprovedanalogously.
PROPOSITION 3.3. Thecanonicalhomomorphism
G →H×L D:={(s ,t )∈H×D|s |R=t |R},
givenbyg 7→(g | ,g | ),isanisomorphism.
M N
Proof. Injectivity follows from the fact thatU =M·N. To establish surjectivity, let s ∈H and t ∈D be such
thats | =t | =:l ∈L .Nowchoosel˜ ∈G suchthatl˜| =l ,anddefineelements
R R R
s ′:=s ◦l˜|−1∈GalD (M/R); and
M
t ′:=t ◦l˜|−1∈GalD (N/R).
N
ByLemma3.2,thereexists˜ ∈GalD (U/N)andt˜∈GalD (U/M)suchthats˜|M =s ′ andt˜|N =t ′.Acomputation
showsthatg :=s˜◦t˜◦l˜ ∈G satisfiesg | =s andg | =t .
M N
COROLLARY 3.4. ThePPV-groupL ={1}ifandonlyifG ≃H×D.Inthiscase,
ea eb a b
G≃ ∈H, e∈D .
(cid:26)(cid:18)ec ed(cid:19)(cid:12)(cid:18)c d(cid:19) (cid:27)
(cid:12)
(cid:12)
WewillnowapplyProposition 3.3tocomputeG(cid:12)incasesI,II,III,andIV.
5
CARLOS E. ARRECHE
4. Explicitcomputations
IncaseI,thereexistsasolutionu∈K fortheRiccatiequationP (u)=0.Wemaychoosethebasis{h ,x }forthe
q
solution spaceof(3.1)suchthatd h =uh andd x =h −2.Theembedding H ֒→SL (F)isthengivenbythe
x x h 2
formulae s (h )=as h and s (x )=a−s 1x +bs h (c(cid:0)f. R(cid:1)emark 3.1), and there are differential algebraic subgroups
A 6 G (F) and B 6 G (F) such that H is defined by (3.3) (see [1,7] for more details). We let Q := P N
m a
denotethefreecommutativemonoidonthesetP (see§2).TheD -fieldL:=Khh iD isaPPV-extension ofK for
d xY−uY =0,andA≃GalD (L/K).Wearereadytostatethemainresultofthissection.
THEOREM 4.1. IncaseI,withnotationasabove,exactlyoneofthefollowingpossibilitiesholds:
(i) Thereexistintegersk ,k ∈Z,withgcd(k ,k )assmallaspossible,suchthattheimagegroup
1 2 1 2
{ak1 |a∈A}=6 {1},
andsuchthatthereexistsanelement f ∈Ksatisfying
k u−k r = d xf.
1 2 1 f
(ii) Case(i)doesn’thold,andthereexistlineardifferentialpolynomialsp,q∈F{Y ,...,Y }1 suchthatthe
1 m P
imagegroup
p ¶ 1a,...,¶ ma a∈A 6={0},
a a
(cid:8) (cid:0) (cid:1)(cid:12) (cid:9)
andsuchthatthereexistsanelement f ∈Ksatisfying(cid:12)
p(¶ u,...,¶ u)−q(¶ r ,...,¶ r )=d f.
1 m 1 1 m 1 x
TheF-vectorspacegeneratedbysuchpairs(p,q)admitsafiniteF-basis{(p ,q ),...,(p ,q )}.
1 1 s s
(iii) Thereexistlineardifferentialpolynomialsp∈F{Y}1 andq∈F{Y ,...,Y }1 suchthattheimagegroup
P 1 m P
{p(b)|b∈B}=6 {0},
andsuchthatthereexistsanelement f ∈Ksatisfying
p(h −2)−q ¶ r ,...,¶ r =d f.
1 1 m 1 x
(cid:0) (cid:1)
TheF-vectorspacegeneratedbysuchpairs(p,q)admitsafiniteF-basis{(p ,q ),...,(p ,q )}.
1 1 s s
(iv) L ={1}.
Consequently,ineachofthesecasesGcoincideswiththesubsetofmatricesin
ea eb
a∈A, b∈B, e∈D (4.1)
(cid:26)(cid:18)0 ea−1(cid:19)(cid:12) (cid:27)
(cid:12)
(cid:12)
thatsatisfythecorrespondingsetofconditionsbelow(cid:12):
(1) Incase(i),ak1 =ek2.
(2) Incase(ii),p ¶ 1a,...,¶ ma =q ¶ 1e,...,¶ me for16i6s.
i a a i e e
(3) Incase(iii),p(cid:0)(b)=q ¶ 1e(cid:1),...,¶(cid:0)me for16i(cid:1)6s.
i i e e
(4) Incase(iv),thereareno(cid:0) furthercon(cid:1)ditions.
We will show slightly more in the course of the proof. We collect these facts in the form of criteria that
eliminate from consideration certain possibilities from Theorem 4.1without testing them directly, based onthe
dataofH andDonly.
6
COMPUTING PPVGROUPS OF SECOND-ORDER EQUATIONS
COROLLARY 4.2. ThefollowingcasesrefertothelistofpossibilitiesinthefirstpartofTheorem4.1.
(1) Incase(i),eitherAandDarebothfinite,orelsetheycoincideassubgroupsofG (F).
m
(2) Incase(ii),neitherAnorDisP -constant.
(3) Incase (iii),A⊆{±1},B6={0},andD isnotP -constant.Itfollowsthath 2 ∈K,andthereforethetest
comprisedin(iii)concernselementsofKonly.
ProofofTheorem4.1. That the possibilities (i)–(iv) are exhaustive and mutually exclusive will be proved in
Propositions 4.4 and 4.6 below. Let us prove that each of these possibilities implies that G is defined as a
subsetof(4.1)bythecorresponding equations contained in(1)–(4),andthatnomoreequations arerequired. In
case(iv),thefactthatGcoincides with(4.1)isCorollary3.4.
In case (i),acomputation showsthat h k1z −k2 =cf forsome c∈F.Letting ag := hgh and eg := zgz for g ∈G ,
weseethatg (cf)=cf impliesthatakg 1 =ekg 2,andthatAisfiniteifandonlyifDisfinite.IfAandDareinfinite,
Theorem 2.3 shows that A and D are defined by the same linear differential polynomials (see Proposition 2.3),
because
k q ¶ 1ag ,...,¶ mag =k q ¶ 1eg ,...,¶ meg (4.2)
1 ag ag 2 eg eg
foreveryg ∈G andeveryq∈F{Y ,...,(cid:0)Y }1,andthe(cid:1)integer(cid:0)sk andk ar(cid:1)ebothdifferent fromzero.Itfollows
1 m P 1 2
from (4.2) that there are no more differential-algebraic relations defining G as a subset of (4.1) which involve
only Aand D.Tosee that there are nomore relations atall, itsuffices toshow that Bisin thekernel ofH ։L
(by Propostition 3.3), or equivalently that R⊆L, which follows from Propositions 4.4 and 4.6. This concludes
theproofof(1).
Incase(ii),acomputation showsthat
¶ h ¶ h ¶ z ¶ z
p 1 ,..., m −q 1 ,..., m ∈K.
h h z z
(cid:0) (cid:1) (cid:0) (cid:1)
Since,foreachg ∈G ,
g p ¶ 1h ,...,¶ mh =p ¶ 1h ,...,¶ mh +p ¶ 1ag ,...,¶ mag ;
h h h h ag ag
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
and
g q ¶ 1z ,...,¶ mz =q ¶ 1z ,...,¶ mz +q ¶ 1eg ,...,¶ meg ,
z z z z eg eg
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
theparameterized Galoiscorrespondence impliesthat
p ¶ 1ag ,...,¶ mag =q ¶ 1eg ,...,¶ meg
ag ag eg eg
(cid:0) (cid:1) (cid:0) (cid:1)
foreachg ∈G .IfA(resp.D)wereP -constant, then ¶ jh (resp. ¶ jz )wouldbelongtoK forevery¶ ∈P ,whichis
h z j
impossible. Inparticular, Aisinfinite, soLemma4.5says thatBisinthekernel ofH ։L .ByProposition 4.6,
incase(ii)
¶ h ¶ h ¶ z ¶ z
K 1 ,..., m ∩K 1 ,..., m =R.
h h D z z D
ByPropostion 3.3,therearenomore(cid:10)equations de(cid:11)fining(cid:10)Gin(4.1).T(cid:11)hisconcludes theproofof(2).
Incase(iii),acomputation showsthat
x ¶ z ¶ z
p −q 1 ,..., m ∈K.
h z z
(cid:0) (cid:1) (cid:0) (cid:1)
Letbg :=g hx −hx foreachg ∈G .Since
g p hx =p hx +p(bg ) and g q ¶ z1z ,...,¶ mz z =q ¶ z1z ,...,¶ mz z +p ¶ e1geg ,...,¶ megeg ,
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
7
CARLOS E. ARRECHE
we see that p hx ∈/ L, and p(bg )=q ¶ e1geg ,...,¶ megeg . Since the element h:= p hx ∈R does not belong to L,
B is not in the(cid:0)ke(cid:1)rnel of H ։L . By L(cid:0)emma 4.5, th(cid:1)is implies that A⊆{±1}. B(cid:0)y(cid:1)Proposition 4.4, in this case
L∩R=K.Therefore, theequations definingGin(4.1)donotinvolve A.Thisconcludes theproofof(3).
In proving Theorem 4.1, itisconvenient totreat separately the cases where A⊆{±1}and A*{±1}. This
is done in Proposition 4.4 and Propostion 4.6, respectively. These results are obtained as consequences of the
Kolchin-Ostrowski Theorem[13].
THEOREM 4.3 (Kolchin-Ostrowski). LetK ⊆V bead x-fieldextensionsuchthatVd x =Kd x,andsupposethat
e ,...,e ,f ,...,f ∈V areelementssuchthatd xei ∈K foreach16i6m,andd f ∈Kforeach16 j6n.
1 m 1 n ei x j
Then,theseelementsarealgebraicallydependentoverKifandonlyifatleastoneofthefollowingholds:
(i) Thereexistintegersk ∈Z,notallzero,suchthat(cid:213) m eki ∈K.
i i=1 i
(ii) Thereexistelementscj ∈Kd x,notallzero,suchthat(cid:229) nj=1cjfj ∈K.
ThefollowingresultimpliesTheorem4.1incaseA⊆{±1}.
PROPOSITION 4.4. IfA⊆{±1},thenh 2∈K andexactlyoneofthefollowingpossibilitiesholds:
(i) A={±1},Disfiniteofevenorder2k,and
u−kr = d xf
1 f
forsome f ∈K.Moreover,inthiscaseR=L.
(iii) Thereexistlineardifferentialpolynomialsp∈F{Y}1 andq∈F{Y ,...,Y }1 suchthattheimagegroup
P 1 m P
{p(b)|b∈B}=6 {0},
andsuchthatthereexistsanelement f ∈Ksatisfying
p(h −2)−q ¶ r ,...,¶ r =d f.
1 1 m 1 x
(cid:0) (cid:1)
TheF-vectorspacegeneratedbysuchpairs(p,q)admitsafiniteF-basis{(p ,q ),...,(p ,q )}.Moreover,
1 1 s s
inthiscaseR∩L=K.
(iv) L ={1}.
Proof. IfA⊆{±1},thensh =±h foreverys ∈H,whenceh 2∈K andL=K(h ).IfL 6={1},thereexistsan
element f ∈R such that f ∈/ K, and there exist non-constant rational functions Q1(Y,Z¶ ,q ) and Q2(Y,Zq ) with
coefficients inK,wherethevariablesareindexedby¶ ∈P andq ∈Q ,suchthat
Q z ,q ¶z = f =Q h ,q x . (4.3)
1 z 2 h
(cid:16) (cid:17) (cid:16) (cid:17)
We may assume that the powers of z (resp. h ) appearing in the numerator and denominator of Q (resp. Q )
1 2
are algebraically independent over K, and that the q ¶z (resp. q x ) appearing in Q (resp. Q ) are F-linearly
z h 1 2
independent. Since d q ¶z ∈K and d q x ∈K for each q ∈Q and ¶ ∈P , Theorem 4.3 implies that these q ¶h
x z x h h
(resp. q x ) are algebraically independent over K(z ) (resp. L). Clearing denominators in (4.3) shows that the
h
elements z ,h , q ¶z , q x are algebraically dependent over K. Since d xz , d xh ∈ K for each ¶ ∈ P and q ∈ Q ,
z h z h
Theorem4.3saysthatthereexistintegersk1,k2∈Z,noneofthemzero,suchthath k1z k2 ∈K,orelsethereexist
cq ∈F,almostallzerobutnotallzero,anddj,q ∈F,almostallzerobutnotallzero,where16 j6m,suchthat
(cid:229) cq q hx −(cid:229) dj,q q ¶ zjz ∈K. (4.4)
q j,q
8
COMPUTING PPVGROUPS OF SECOND-ORDER EQUATIONS
First suppose that h k1z k2 ∈K as above. Since L 6={1}, we may assume that k1 =1, h ∈/ K, and therefore
z k2 ∈/ K. Now hz k2 ∈K implies that z 2k2 ∈K, whence D is finite. Let k >0 be the smallest integer k2. Then
K(z k)=L=R,theorderofDis2k,and
u−kr = d x(zh −k) = d xf.
1 zh −k f
Supposing insteadthatthereareelementscq ,d¶ ,q ∈F asin(4.4),set
p:=(cid:229) cq q Y ∈F{Y}1P ; and q:=(cid:229) dj,q q Yj ∈F{Y1,...,Ym}1P ,
q j,q
sothat(4.4)nowreads
x ¶ z ¶ z
p −q 1 ,..., m =: f ∈K.
h z z
(cid:0) (cid:1) (cid:0) (cid:1)
SinceL 6={1},wemayassumethatp x ∈/ K,andthat
h
(cid:0) (cid:1)
¶ z ¶ z
q 1 ,..., m ∈/K.
z z
(cid:0) (cid:1)
¶z
ThisimpliesthatDisinfinite,andhenceconnected[4,15],becausewheneverDisfinitewehavethat ∈K for
z
each¶ ∈P .Letbg :=g hx −hx andeg := zgz foreachg ∈G ,andnotethat
g p hx =p hx +p(bg ).
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1)
Hence,thereexistsb∈Bsuchthatp(b)6=0.Observethatd p x =p(h −2);and
x h
(cid:0) (cid:1)
d q ¶ 1z ,...,¶ mz =q ¶ r ,...,¶ r .
x z z 1 1 m 1
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1)
Thefinite-dimensionality oftheF-vectorspacegenerated bysuchpairs(p,q)followsfromthefactthatbothN
and M have finite algebraic transcendence degree over K (since H/R (H)≃A⊆{±1} is P -constant [19]), so
u
weonlyneedtoconsider finitelymanyelementsfrom{q x ,q ¶z |¶ ∈P ,q ∈Q }.
h z
Wenowconsider thecasewhenA*{±1}.Webeginwithapreliminary result.
LEMMA 4.5. IfA*{±1},thenBisinthekerneloftherestrictionhomomorphismH ։L .
Proof. ToshowthatR⊆L,weproceedbycontradiction:supposethat f ∈Rand f ∈/L.Thereexistnon-constant
rational functions Q1(Y,Z¶ ,q ) and Q2(Zq ) with coefficients in L, where the variables are indexed by q ∈Q and
¶ ∈P ,suchthat
Q z ,q ¶z = f =Q q x . (4.5)
1 z 2 h
Weassume without loss ofgenerality that the(cid:0)q ¶z (r(cid:1)esp. q x )ap(cid:0)pear(cid:1)ing inQ (resp. Q )are algebraically inde-
z h 1 2
pendent over L. Clearing denominators in (4.5) shows that the elements z ,q ¶z ,q x are algebraically dependent
z h
overL.Since
d xz , d q ¶z , d q x ∈L
z x z x h
foreachq ∈Q and¶ ∈P ,andsince f ∈/ L,byTheorem 4.3thereexist cq ∈F =Ld x,almost allzero butnotall
zero,andd¶ ,q ∈F,almostallzerobutnotallzero,suchthat
(cid:229) cq q hx + (cid:229) d¶ ,q q ¶zz =:g∈L. (4.6)
q ∈Q ¶ ∈P ,q ∈Q
9
CARLOS E. ARRECHE
Applyingd onbothsidesof(4.6),weobtain
x
(cid:229) cq q (h −2)+ (cid:229) d¶ ,q ¶q r1=d xg. (4.7)
q ∈Q ¶ ∈P ,q ∈Q
Now choose s ∈GalD (L/K) such that as := hsh ∈FP and a2s 6=1, and apply (s −1) to both sides of (4.7), to
obtain
(a−s 2−1) (cid:229) cq q (h −2)=d x(s g−g).
q ∈Q
Butthisimpliesthat(cid:229) cq q hx ∈L,acontradiction. Thisconcludes theproofthatR⊆L.
PROPOSITION 4.6. IfA*{±1},thenR⊆L,andexactlyoneofthefollowingpossibilitiesholds:
(i) Thereexistintegersk ,k ∈Z,withgcd(k ,k )assmallaspossible,suchthattheimagegroup
1 2 1 2
{ak1 |a∈A}=6 {0},
andsuchthatthereexistsanelement f ∈Ksatisfying
k u−k r = d xf.
1 2 1 f
(ii) Case(i) doesn’thold,andthereexistlineardifferentialpolynomialsp,q∈F{Y ,...,Y }1 suchthatthe
1 m P
imagegroup
p ¶ 1a,...,¶ ma a∈A 6={0},
a a
(cid:8) (cid:0) (cid:1)(cid:12) (cid:9)
(cid:12)
andsuchthatthereexistsanelement f ∈Ksatisfying
p(¶ u,...,¶ u)−q(¶ r ,...,¶ r )=d f.
1 m 1 1 m 1 x
TheF-vectorspacegeneratedbysuchpairs(p,q)admitsafiniteF-basis{(p ,q ),...,(p ,q )}.Moreover,
1 1 s s
inthiscase
¶ h ¶ h ¶ z ¶ z
K 1 ,..., m ∩K 1 ,..., m =R. (4.8)
h h D z z D
(iv) L ={1}. (cid:10) (cid:11) (cid:10) (cid:11)
Proof. SinceA*{±1},Lemma4.5saysthatR⊆L.AssumethatL 6={1},andlet f ∈Rsuchthat f ∈/K.Then
thereexistnon-constantrationalfunctionsQ1(Y,Z¶ ,q )andQ2(Y,Z¶ ,q )withcoefficientsinK,wherethevariables
areindexedby¶ ∈P andq ∈Q ,suchthat
Q (h ,q ¶h )= f =Q (z ,q ¶z ). (4.9)
1 h 2 z
We assume without loss of generality that the q ¶h (resp. q ¶z ) appearing in Q (resp. Q ) are algebraically
h z 1 2
independent over K (cf. the proof of Proposition 4.4). Clearing denominators in (4.9) shows that the elements
h , z , q ¶h , q ¶z arealgebraically dependent overK.Since
h z
d xh , d xz , d q ¶h , d q ¶z ∈K;
h z x h x z
Theorem 4.3 implies that either there are integers k1,k2 ∈Z, none of them zero, such that h k1z −k2 ∈K, or else
thereexistcj,q ∈F,almostallzerobutnotallzero,anddj,q ∈F,almostallzerobutnotallzero,where16 j6m,
suchthat
(cid:229) cj,q q ¶ hjh −(cid:229) dj,q q ¶ zjz ∈K. (4.10)
j,q j,q
10
