Shift Equivalence of P-finite Sequences(9)

We present an algorithm which decides the shift equivalence problem for Pfinite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial coefficients. Two sequences are called shift equivalent if shi

4

5

6whileifr:=r 1else//nowvr=0

ifreturn

10

11r=1then{s∈:ur/vr=αs}s:=α(ur 1/ur vr 1/vr)

ifu=Jsvthen{s}else

ThecorrectnessofAlgorithms1and2shouldbeclearbytheabovediscussion.Severalrestrictions,however,havetobemadeforthe eldkinorderthateverystepinthesealgorithmscanbecarriedoutalgorithmically.Ofcourse,itisnecessarythatkisacom-putable,i.e.,thateveryelementhasa niterepresentation,thatthearithmeticoperations+, ,·,/arecomputable,andthatzeroequivalencecanbedecided.Furthermore,forthecomputationofaJordandecomposition(Line2inAlg.1),weneedtobeabletocompute¯alsohasabsolutefactorizationsofunivariatepolynomialsink[X].Thealgebraicclosurek

tobeacomputable eld.Line11ofAlgorithm2requirestodecidewhetheranelement¯isaninteger.Alltheserequirementscanbeaccommodatedformost eldskthatofk

mightbeofinterest.Morerestrictiveisthe nalrequirement,originatingfromline9:We¯.Analgorithmhavetobeabletocomputetheset{s∈:a=bs}forgivena,b∈k

forthispurposewasgivenbyAbramovandBronstein[1].Thisalgorithmisapplicablewheneverkissuchthatitcanbedecidedforanygivenx∈kwhetherxistranscendentaloralgebraicover,andthatforanytwoelementsx,y∈kitcanbedecidedwhethertheseelementsarealgebraicallyindependentover.Ge’salgorithm[8]givesrisetoane cientalternativeifkisasinglealgebraicextensionof,i.e.,ifk=(α)forsomealgebraicnumberα.

3.3Summary

Lemma2reducestheshiftequivalenceproblemforC- nitesequencetosolvingamatrixequation,andthismatrixequationcanbesolvedbymeansofAlgorithm1.Puttingthingstogether,wethusobtainthefollowingalgorithmforsolvingtheshiftequivalenceproblemforC- nitesequences.

Algorithm3

INPUT:f1,f2:→kC- nite,speci edbyannihilatingoperatorsL1,L2∈k[E]andinitialvalues

OUTPUT:alls∈suchthatf1=Esf2

1function

L·f1=0orreturn

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