A Refined Denominator Bounding Algorithm for Multivariate Linear Difference Equations

0.19 MB·English

by  Manuel Kauers

A Refined Denominator Bounding Algorithm for
Multivariate Linear Difference Equations
∗ †
Manuel Kauers Carsten Schneider
RISC RISC
JohannesKeplerUniversity JohannesKeplerUniversity
4040Linz(Austria) 4040Linz(Austria)
[email protected] [email protected]
1
1
0 ABSTRACT The first algorithm for finding a denominator bound Q
2 was given by Abramov in 1971 [2, 3, 4]. During the past
We continue to investigate which polynomials can possibly
fourty years, other algorithms were found [12, 15, 6, 8] and
n occurasfactors inthedenominatorsofrationalsolutionsof
a a given partial linear differenceequation. In an earlier arti- the technique was generalized to matrix equations [1, 5] as
J well as to equation over function fields [13, 7, 14]. Last
cle we had introduced the distinction between periodic and
year[11]wemadeafirststeptowardsadenominatorbound-
4 aperiodic factors in the denominator, and we gave an algo-
1 rithmforpredictingtheaperiodicones. Nowweextendthis ing algorithm for equations in several variables (PLDEs).
technique towards the periodic case and present a refined We found that some factors of the denominator are easier

] algorithm which also findsmost of theperiodic factors. to predict than others. We called a polynomial periodic if
C it has a nontrivial gcd with one of its shifts, and aperiodic
S Categories andSubject Descriptors otherwise. For example, the polynomial 2n−3k is periodic
s. I.1.2 [Computing Methodologies]: Symbolic and Alge- because shifting it twice in k and three times in n leaves
it fixed. We say that it is periodic in direction (3,2). An
c braic Manipulation—Algorithms
[ example for an aperiodic polynomial is nk+1. The main
General Terms result oflast year’s paperwasan algorithm for determining

1 aperiodic denominator bounds for PLDEs, i.e., we can find
v Algorithms Qsuchthatwhenevery= p solvesthegivenequationand
3 Keywords q is aperiodic, then q|Q. uq
0
The present paper is a continuation of this work. We
8 Difference Equations,Rational Solutions
nowturntoperiodicfactorsandstudyunderwhichcircum-
2
stancesaslightlyadaptedversionoflastyear’salgorithmcan
. 1. INTRODUCTION
1 also predict periodic factors of the denominator. We pro-
0 The usual approach for finding rational solutions of lin- poseanalgorithmwhichfindstheperiodicfactorsforalmost
1 ear difference equations with polynomial coefficients is as all directions. Every equation has however some directions
1 follows. First one constructs a nonzero polynomial Q such which our algorithm does not cover. But if, for instance,
v: thatforanysolutiony=p/q ofthegivenequationwemust wehaveasystem oftwoequationsandapplyouralgorithm
i have q | Q. Such a polynomial Q is called a denominator to each of them, then the two bounds can under favorable
X bound for the equation. Next, the denominator bound is circumstances(whichcanbedetectedalgorithmically)com-
r used to transform the given equation into a new equation bined to a denominator bound which provably contains all
a withthepropertythatapolynomialP solvesthenewequa- the factors that can possibly occur in the denominator of
tion if and only if the rational function y =P/Q solves the anysolutionofthesystem. Thiswasnotpossiblebefore. So
original equation. Thus the knowledge of a denominator while until now we were just able to compute in all situa-
bound reducesrational solving topolynomial solving. tions some factors, we can now also find in some situations
all factors.
∗
Supportedby the Austrian FWF grant Y464-N18 and the
Despite this progress, we must confess that our results
EU grant PITN-GA-2010-264564.
are still of asomewhat academic naturebecausedenomina-
†
Supported by the Austrian FWF grant P20347-N18 and tor boundsin which some factors are missing are not really
theEU grant PITN-GA-2010-264564.
enoughforsolvingequations. Andevenwhenafulldenomi-
natorboundisknown,itstillremainstofindthepolynomial
solutionsofaPLDE,andnobodyknowshowtodothis—the
correspondingproblemfor differentialequationsisundecid-
able. But in practice, we can heuristically choose a degree
Permission tomake digital orhardcopies ofall orpartofthis workfor
personalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesare boundforfindingpolynomialsolutions,andknowingpartsof
notmadeordistributedforprofitorcommercialadvantageandthatcopies the possible denominators is certainly better than knowing
bearthisnoticeandthefullcitationonthefirstpage.Tocopyotherwise,to nothing, and the more factors we know, the better. Apart
republish,topostonserversortoredistributetolists,requirespriorspecific from this, we find it interesting to see how far the classi-
permissionand/orafee.
cal univariate techniques carry in the multivariate setting,
Copyright200XACMX-XXXXX-XX-X/XX/XX...$5.00.
and we would be curious to see new ideas leading towards 2. Ifpandq areirreducible,thenthereexistss∈ r such
algorithms which also find thefactors that we still miss. that s+Spread(p,q) is a submodule of r. Z
Z
3. There is an algorithm for computing Spread(p,q).
2. PREPARATIONS
Let bea field of characteristic zero. We considerpoly- 4. There is an algorithm for computing an aperiodic de-
nomialKs and rational functions in the r variables n ,...,n nominator bound for (1) given the support S and the
1 r
with coefficients in . For each variable ni, let Ni denote coefficients as (s∈S).
K
theshiftoperatormappingn ton +1andleavingallother
i i
variables fixed,so that 3. DENOMINATOR BOUNDS MODULO A
PRESCRIBED MODULE
N q(n ,...,n )
i 1 r
=q(n ,...,n ,n +1,n ,...,n ) Ourgoal inthissection istodeterminethefactors whose
1 i−1 i i+1 r
spread is contained in some prescribed set W ⊆ r. Under
for every rational function q. Whenever it seems appropri- suitable assumptions about W such factors mustZpop up in
ate, we will use multiindex notation, writing for instance n the coefficients of the equation (cf. Lemma 2 below) and
instead of n1,...,nr or Ni for N1i1N2i2···Nrir. under stronger assumptions we can also give a bound on
Weconsider equations of theform the dispersion between them (cf. Theorem 2 below). Using
these two results we obtain a denominator bound relative
XasNsy=f (1) to W (cf. Theorem 3 and Algorithm 1) below. In the next
s∈S section, we then propose an algorithm which combines the
where S ⊆ r is finite and nonempty, f ∈ [n] and as ∈ denominatorboundswithrespecttoseveralsetsW. Itturns
Z K
[n]\{0} (s∈S) are given, and y is an unknown rational out that by considering only finitely many sets W one can
K
function. Ourgoalistodeterminethepolynomialsp∈ [n] obtainadenominatorboundwithrespecttoinfinitelymany
K
whichmaypossiblyoccurinthedenominatorofasolutiony, sets W.
or at least to findmany factors of p.
We recall the following definitions and results from our Definition 2. Let W ⊆ r with 0 ∈ W. A polynomial
Z
previous paper[11]. d ∈ [n]\{0} is called a denominator bound of (1) with
K
respect to W if for every solution y = p/q ∈ (n) of (1)
Definition 1. Let p,q,d∈ [n]. and every irreducible factor u of q with SpreadK(u)⊆W we
K
have u|d.
1. The set Spread(p,q) := {i ∈ r : gcd(p,Niq) 6= 1}
Z
is called the spread of p and q. For short, we write Typically,W willbeasubmoduleof r oraunionofsuch
Spread(p):=Spread(p,p). modules. The definition reduces to theZnotion of aperiodic
denominator bound when W = {0}. In the other extreme,
2. The number Dispk(p,q) := max{|ik| : (i1,...,ir) ∈ when W = r then d is a“complete”denominator bound:
Spread(p,q)} is called the dispersion of p and q with Z
itcontainsallthefactors, periodicornot,thatcanpossibly
respect to k ∈{1,...,r}. (We set maxA:=−∞ if A
occur in the denominator of a solution y of (1). In general,
is empty and maxA:=∞ if A is unbounded.)
dpredictsallaperiodicfactorsinthedenominatorofasolu-
tionaswellastheperiodicfactorswhosespreadiscontained
3. The polynomial p is called aperiodic if Spread(p) is
in W.
finite, and aperiodic otherwise.
Denominatorboundswithrespecttodifferentsubmodules
4. The polynomial d is called an aperiodic denominator can becombined as follows.
bound for equation (1) if d 6= 0 and every solution y
can be written as a for some a,b,u ∈ [n] where u Lemma 1. Let W1,...,Wm be submodules of r, and let
is periodic and b|du.b K d1,...,dm be denominator bounds of (1) withZrespect to
W ,...,W , respectively. Then d := lcm(d ,...,d ) is a
1 m 1 m
5. Apointp∈S ⊆ r ⊆ r iscalleda cornerpointofS denominator bound with respect to W :=W ∪···∪W .
ifthere exists aveZctor vR∈ r suchthat (s−p)·v>0 1 m
R Proof. Letubeanirreduciblefactorofthedenominator
for all s ∈ S\{p}. Such a vector v is then called an
innervector,andtheaffinehyperplaneH :={x∈ r : of some solution of (1) and suppose that U :=Spread(u)⊆
(x−p)·v=0} is called a borderplane for S. R W. ItsufficestoshowthatthenU ⊆Wkforsomek,because
then it follows that u|d |d,as desired.
k
We show that if U contains some vector x 6∈ W , then
1
W ⊆W ∪···∪W ,hence U ⊆W ∪···∪W . Applying
1 2 m 2 m
theargument repeatedly provesthat U ⊆W for some k.
k
Let y ∈ W . Since U is a submodule of r, we have
1
Z
x+αy∈U forallα∈ . ByassumptionU ⊆W ∪···∪W ,
1 m
Z
soeachsuchx+αy mustbelongtoatleastonemoduleW
ℓ
(ℓ = 1,...,m). It cannot belong to W though, because
1
together with y ∈ W this would imply x ∈ W , which
1 1
Theorem 1. Let p,q∈ [n]. is not the case. Therefore: For every α ∈ there exists
K ℓ∈{2,...,m} such that x+αy∈W . Z
ℓ
1. Ifpisirreducible,thenSpread(p)isasubmoduleof r Since isinfiniteandmisfinite,theremustbesomeindex
Z Z
and p is aperiodic if and only if Spread(p)={0}. ℓ ∈{2,...,m} for which there are two different α ,α ∈
1 2
Z
with x+α y ∈ W and x+α y ∈ W . Since W is also Conversely, the factor 4k − 2n+ 1 cannot possibly ap-
1 ℓ 2 ℓ ℓ
a submodule of r, it follows that (α −α )y ∈ W , and pear in the denominator of a solution, because for W′ :=
1 2 ℓ
finally y∈W ⊆ZW ∪···∪W ,as claimed. Spread(4k − 2n + 1) = 1 we can take p′ = 1 and
ℓ 2 m (cid:0)2(cid:1)Z (cid:0)0(cid:1)
v′ = −1 , and according to the lemma, some shifted ver-
Thenextresult saysthatfactorsofdenominatorstendto (cid:0)1/2(cid:1)
sion of 4k−2n+1 would have to appear in the coefficient
leave traces in thecoefficients of corner points of S.
of y(n+1,k).
Lemma 2. Let W be a submodule of r and let u with More generally, for any nontrivial submodule W′′ of 2
Spread(u)⊆W be an irreducible factor oZf the denominator other than W, Lemma 2 excludes the possibility of periodZic
of some solution y of (1). Let p ∈ S be a corner point of factors whose spread is contained in W′′, because such fac-
S with an inner vector v ∈ r orthogonal to W (meaning tors would have to leave a trace in at least one of the coeffi-
w·v = 0 for all w ∈ W). RThen there exists i ∈ r such cients of the equation.
that Niu|ap. Z
Proof. If u′ is another irreducible factor of the denom-
inator of y and Spread(u,u′) is nonempty, then we have
Spread(u,u′) ⊆ c+W for some c ∈ r. This follows from
Z
Theorem 1.2 and the assumption Spread(u) ⊆ W. For the
full denominator d of y, we can thus find c ,...,c ∈ r
1 m
withSpread(u,d)⊆ m (c +W)whereeachelementfroZm
Sk=1 k
C is necessary. Let C ={c ,...,c } be such a choice, and
1 m
let i∈{1,...,m}be such that c ·v is minimal.
i
Wehave In the previous example, we could thusdetermine all the
apNpy=f − X asNsy tinhteerfaescttionrgsmofotdhuelecsoeWfficbieynjtussotflothokeienqguaattitohne.sTphreeafdoslloowftinhge
s∈S\{p}
example indicates that thisis not always sufficient.
as an identity in (n). Therefore, every factor in the de-
nominator of NpyKmust either be canceled by ap or it also
occurs as a factor in at least one of the Nsy (s∈S\{p}). Example 2. The equation
The factor Np+ciu appears in the denominator of Npy.
If it also appeared in the denominator of Nsy for some (2k−3n2−8n−5)y(n,k+1)
s ∈ S\{p}, then this would imply Np+ciu = Ns+cju for
+(k+3n2+5n+4)y(n+1,k)
some j ∈{1,...,m}. But then s−p+c −c ∈W, which
j i
is in contradiction to −(5k−3n2−11n−7)y(n+1,k+1)
(s−p+c −c )·v=(s−p)·v+(c −c )·v6=0 +(2k−3n2−8n−3)y(n+2,k)=0
j i j i
| >{z0 } | ≥{z0 } alsohas the solution y=(n2+2k2)/(k+n+1). Its denom-
becauseW isorthogonaltovbyassumption. HenceNp+ciu inator k+n+1 does not appear inany of the coefficients of
cannotappearasadenominatorontherighthandside,and theequation. ThisisbecauseforitsspreadW = 1 there
henceit mustbecanceledon theleft handside. Thisforces are no suitable p ∈ S and v ∈ 2 matching the(cid:0)−co1(cid:1)nZditions
Np+ciu|ap, so theclaim is provenfor i:=p+ci. ofthelemmabecause thepointsR1 , 0 aswellasthepoints
(cid:0)0(cid:1) (cid:0)1(cid:1)
The lemma tells us for which choices of W ⊆ r some- 2 , 1 lie on a line parallel to S.
Z (cid:0)0(cid:1) (cid:0)1(cid:1)
thing nontrivial may happen. Let us illustrate this with an
example.
In summary, in order for W to be the spread of a factor
Example 1. The equation that can appear in the denominator of a solution of (1),
W mustbecontainedinthespreadofsomecoefficientofthe
(4k−2n+1)(k+n+1)y(n,k)
equation (as in Ex. 1) or it must be parallel to one of the
+(8k2+2kn+k+6n2+13n+6)y(n,k+1) faces in the convex hull of the support S (as in Ex. 2). For
every equation, we can thus determine some finitely many
−2(6k2+2kn+13k+2n2+n+6)y(n+1,k)=0 submodulesof rofcodimensiononesuchthateachpossibly
Z
has the solution y = (n2 +2k2)/(k+n+1). Its denom- occuring spread W is contained in at least one of them.
inator is periodic, Spread(k+n+1) = 1 . Lemma 2
predicts the appearance of k+n+1(orat(cid:0)l−ea1s(cid:1)tZsomeshifted 3.1 A NormalizingChange ofVariables
version of it) in the coefficient of y(n,k), because for the Let r = V ⊕ W be a decomposition of r into sub-
choice W = 1 , the point p= 0 ∈S admits the choice moduleZs. Our goal is to obtain denominatorZbounds with
(cid:0)−1(cid:1)Z (cid:0)0(cid:1)
v= 1 in accordance with the requirements imposed by the respect to W by applying the algorithm from last year [11]
lemm(cid:0)a1(cid:1). Notethatnoshiftequivalentcopyofk+n+1occurs to V ∼= r/W. It turns out that this can be done provided
Z
in the coefficients of y(n,k+1) or y(n+1,k), which does that W is sufficiently nondegenerate. In order to formulate
not contradict the lemma, because the points 0 and 1 lie the precise conditions on W without too much notational
(cid:0)1(cid:1) (cid:0)0(cid:1)
on a line parallel to W. This has the consequence that for overhead, it seems convenient to make a change of coordi-
thesepoints, theredoesnotexistavectorv withtherequired nates.
property. Let invertible matrices A = ((a ))r ∈ r×r act on
i,j i,j=1 Q
(n) via Theorem 2. Let
K
A·y(n1,...,nr):=y(cid:0)a1,1n1+a1,2n2+···+a1,rnr, A={(s1,...,sr)∈S :s1 =0}, (2)
B={(s ,...,s )∈S :s =k}.
a n +a n +···+a n , 1 r 1
2,1 1 2,2 2 2,r r
. Suppose that no two elements of A agree in the first t co-
.
. ordinates, and that the same is true for B. Let a′ be those
i
ar,1n1+ar,2n2+···+ar,rnr(cid:1). Spoplryenaodm(fia)l⊆swWhi.chLceotntainallirreduciblefactorsf ofai with
WeobviouslyhaveA·(p+q)=(A·p)+(A·q)andA·(pq)=
(A·p)(A·q) for all p,q ∈ (n). It can be checked that we s:=max{Disp1(a′s,N1−ka′t):s∈A and t∈B}.
K
also have
Then for any solution y =p/q ∈ (n) of (1) and any irre-
A·(Nsy)=NA−1s(A·y) ducible factors u,v of q with SpreKad(u),Spread(v)⊆W we
have Disp (u,v)≤s.
for every A ∈ r×r with |detA|= 1 and every s ∈ r and 1
every y ∈ (nZ). It follows that y ∈ (n) is a soluZtion of Proof. As S is not empty, A,B are nonempty. W.l.o.g.
(1)ifandoKnlyify˜=A−1·y isasolutioKnofthetransformed wemayassumethattheminimalelementofAw.r.t.lexico-
equation graphic order is thezero vector.
Suppose that there are irreducible factors u,v of q with
X(A·as)NA−1sy˜=A·f, Spread(u) ⊆ W, Spread(v) ⊆ W and d := Disp1(u,v) such
s∈S that d>s; takesuch u,v such that d is maximal. Consider
all the factors Nuu and Nvv occurring in q where the first
or equivalentlyof
entryinuandv is0. NotethatbyLemma3thereareonly
Xa˜sNsy˜=f˜, finitely many choices of the first t components, so we can
s∈S˜ choosetwosuchfactorsfromq wherethefirsttcomponents
of u are minimal and the first t components of v are maxi-
where S˜ = {A−1s : s ∈ S}, a˜s := A·aAs (s ∈ S˜), and mal w.r.t. lexicographic order; these factors are denoted by
f˜=A·f. u′,v′ respectively.
Now take A ∈ r with |detA| = 1 such that the first t • First suppose that u′ divides one of the polynomials as
Z
rows of A form a basis of V and the last r−t rows of A with s∈A. In this case we choose the polynomial aw with
form a basis of W. Then the transformation just described w = (w ,...,w ) ∈ B such that (w ,...,w ) is maximal
1 r 2 t
maps the basis vectors of V to the first t unit vectors and w.r.t. lexicographic order (uniqueness is guaranteed by the
the basis vectors of W to the last r −t unit vectors. In assumptionthatnotwoelementsfrom B agreeinthefirstt
other words, we can assume without loss of generality that components). Wecan write (1) in theform
V itselfisgeneratedbythefirsttunitvectorsandW bythe
1
flarostmrn−owt uonni,tuvnecletsosrsoitnheZrwr.isWe estwatielldm. aNkoetethtihsaatsstuhmispctoionn- Nwy= aw(cid:16)f − X asNsy(cid:17). (3)
s∈S\{w}
vention implies for an irreducible polynomial u∈ [n] that
Spread(u)=W isequivalenttoubeingfreeoftheKvariables Now observe that the factor Nwv′ does not occur in the
n ,n ,...,n andaperiodicaselementof [n ,...,n ]. denominator of any Nsy with s∈S\{w}:
t+1 t+2 r 1 t
K
By applying,ifnecessary,a suitablepowerof N onboth
1 1. Supposethatthereiss∈S\B suchthatNwv′ occurs
sides of the equation we can further assume without loss of
in Nsq, i.e., Nw−sv′ is a factor of q. Since the first
generality that min{s : (s ,...,s ) ∈ S} = 0, and we set
1 1 r componentofwisk (w∈B)andthefirstcomponent
k:=max{s :(s ,...,s )∈S}.
1 1 r of s is smaller than k (s∈/ B), the first component of
3.2 Bounding theDispersion w−sispositive. Moreover,sincethedistancebetween
the factors v′ and u′ of q is d in the first component,
With this transformation w.r.t. the submodule W of r the factors v′ and Nw−sv′ of q have distance larger
Z
and under the assumption that the extreme points (2) of
thandinthefirstcomponent;acontradictionthatthe
S havecertain properties, Theorem 2 explainshow one can distancedismaximallychosen. Consequently,ifNwv′
bound the dispersion along the x1-coordinate of all factors is a factor in the denominator of Nsy with s ∈ S, it
f with Spread(f) ∈ W that occur in the denominator of
follows that s∈B.
a solution of (1). This result is a refinement of Lemma 2
from [11]. 2. Suppose that there is s ∈ B with w 6= s such that
Nwv′ is a factor of Nsq. Then Nw−sv′ is a factor of
Lemma 3. Letu,v∈ [n]\{0} withSpread(u)⊆W and q. Since the first component of the vectors in B is
K
Spread(v)⊆W. Then k, but the first t components in total cannot be the
samefor twodifferentvectorsofB,it follows thatthe
|{(s1,...,st)∈ t |∃(st+1,...,sr)∈ r−t: first entry in w −s is zero and at least one of the
Z Z
N(s1,...,sr)u=v}|≤1. others is non-zero; in particular, by the maximality
assumption on w the first non-zero entry is positive.
Proof. Takes,s′withNsu=v=Ns′u. AsNs−s′u=u, Hence we find v′ = (0,v′,...,v′) := v+w−s such
2 r
it follows s−s′ ∈ W = {0}t × r−t, and thus the first t that Nv′v is a factor of q and suchthat (v′,...,v′) is
Z 2 t
components of s,s agree. largerthan(v ,...,v )w.r.t.lexicographicordering;a
2 t
contradiction tothechoice of thevectorv.
Sincef,as∈ [n],thecommondenominatoroftherational Lemma 4. Letpbeacorner pointof S withborder plane
function f −KPs∈S\{w}asNsy does not contain the factor H andinnervectorv. Thenforeverys>0thereexistfinite
Nwv′. Now suppose that Nwv′ is a factor of aw. Since sets
w ∈ B, its first component is k. But then, since u′ and R− ⊆ r∩ (H+ev) and
v′ havedistancedinthefirstcoordinate,alsothefactorsu′ Z [
wanhdicNh1−imkNpliwesv′thhaavtesd≥istda;nacecodn.tTrahduicstDioinsp.1O(avse,rNal1l−,ktahwe)co≥md- R+ ⊆ r∩0≤e(≤Hs +ev),
Z [
mon denominator on theright handsideof (3) cannot con- e>s
tNaiwnythisenfaocttdorivNisiwbvle′bwyhiNchwivm′.plTiehsutshtahtethdeendoemnoinmaitnoartoorf oyf, an(dn)poolfyn(o1m)iwaelshba,vbei ∈K[n] such that for any solution y ∈
in particular q is not divisibleby v′; a contradiction. K
• Conversely, suppose that u′ does not divide any of the Npy= b+Pi∈R+biNiy. (4)
spuoclyhntohmatia(lwsa,s.w..i,twhs)∈isAm.inNimowallewt.rw.t.=le(x0ic,owg2r.a.p.h,iwcro)rd∈eAr- Qi∈R−Ni−pap
2 t
ing (again it is uniquely determined by the assumptions on ThesetsR−andR+andthepolynomialsb,bicanbecom-
A),andwrite(1)intheform (3);byourassumption stated puted for a given s, S, p, and v by Algorithm 2 from [11].
in the beginning, w is just thezero vector 0. By analogous The next theorem provides a denominator bound with re-
arguments as above (the roles of A and B are exchanged) specttoW. ItisanadaptionofTheorem4from[11]tothe
it follows that u′ does not occur in the denominator of any presentsituation. Wecontinuetoassumethenormalization
Nsy withs∈S\{0}. Henceasabove,thecommondenom- V = t×{0}r−t, W ={0}t× r−t.
iun′a.toMroorfeofve−r,Psins∈cSe\u{0′}daoseNssnyotdodeivsidneotancyonatsainfrotmhesfa∈ctAor, ThZeorem 3. Lets∈N∪{−Z∞}besuchthatforanysolu-
the factor u′ does not occur in a0. In total, the factor u′ is tion y =p/q ∈K(n) of (1) and any irreducible factors u,v
not part of the denominator on the right hand side of (3), of q with Spread(u),Spread(v) ⊆ W we have Disp1(u,v) ≤
but it is a factor of the denominator on the left hand side; s. Let p be a corner point of S for which there is an inner
a contradiction. vector v=(v1,...,vr) with v1 ≥1 as well as an inner vec-
torv′ orthogonal toW. Forthesechoicesofs,p,andv,let
Iftherequiredpropertiesonthesets(2)inTheorem2are R−, R+, b, bi be as in Lemma 4. Let a′p be the polynomial
violated, our bounding strategy does not work, as can be consisting of all the factors of ap whose spread is contained
seen by thefollowing example. in W. Then
d:= Ns−2pa′ (5)
Example 3. Fix W := Spread(k+n+1) = 1 and Y p
(cid:0)−1(cid:1)Z s∈R−
take V = 0 . The problem from Example 1 is normalized
(cid:0)1(cid:1)Z is a denominator bound of (1) with respect to W.
bythechangeofvariablesn→k andk→n−k (i.e.,abasis
transformation 0 1 with determinant −1 is chosen) and Proof. Let y = p/q ∈ (n) be a solution of (1) and
(cid:18)1 −1(cid:19) K
one obtains V′ = 1 and W′ = 0 . This gives the new let u be an irreducible factor of q with multiplicity m and
equation (cid:0)0(cid:1)Z (cid:0)1(cid:1)Z Spread(u)⊆W. Wehavetoshowum |d. Lemma2applied
topandv′ impliesthatthereissomei∈ r withu′ |qand
(n+1)(−6k+4n+1)y(n,k) u′ :=Niu|ap. BythechoiceofswehaveZDisp1(u′,u)≤s.
+(12k2−14nk+12k+8n2+n+6)y(n+1,k) Lemma 4 implies therepresentation
−2(6k2−10nk−12k+6n2+13n+6)y(n+1,k+1)=0 Npy= b+Pi∈R+biNiy.
with the new structure set S′ = { 0 , 1 , 1 } which now Qi∈R−Ni−pap
has the solution y = 3k2−4nk+2n2 (cid:0)w0(cid:1)he(cid:0)r0e(cid:1)th(cid:0)1e(cid:1)denominator Because of v1 > 1, every i ∈ R+ differs from p in the first
n+1 coordinate by more than s. This implies that Npu and
consists of the factor n+1 with Spread(n+1) = W′. As hencethatNpum cannotappearinthedenominatorofNiy
observedalreadyinExample1onecanpredictthefactorn+1 for any i ∈ R+. But it does appear in the denominator
(up to a shift in n) by exploiting Lemma 2. However, one of Npy, so it must appear as well in the denominator of
cannot apply Theorem 2. For S′ we get the sets A ={ 0 }
(cid:0)0(cid:1) the right hand side. The only remaining possibility is thus
ianndthBe fi=rs{t(cid:0)c10o(cid:1),m(cid:0)p11o(cid:1)}newnhtebruetindiffBerthienttwhoevseeccotonrdscaormepthoenesnatm.e Npum |Qi∈R−Ni−pap, and hence
3.3 Denominator Bounding Theorem um | Y Ni−2pap.
i∈R−
Thedenominatorboundingtheoremsaysthatifwerewrite Because of Spread(u′) = Spread(u) ⊆ W, it follows that
theequation(1)intoanewequationwhosesupportcontains um |d.
somepointpwhichissufficientlyfarawayfromalltheother
points in the support, then we can read off a denominator Thefollowingfigureillustratesthesituation. Thevectorv
bound from this new equation. We will need the following isorthogonaltoHbutnotnecessarilytoW,whilethevector
fact,whichappearsliterallyasTheorem3in[11](withW,S′ v′isorthogonaltoW butnotnecessarilytoH. Relation(4)
renamedtoR−,R+hereinordertoavoidanameclashwith separates p from the points in R+ which are all below the
themeaning of W in thepresent paper). plane H +sv. The points in R− are all between H and
H+sv.
10 Choose an inner vector v∈ r for p.
11 Compute R− as defined in LRemma 4.
12 Compute d as defined in Theorem 3.
13 Apply the inverse change of variables to d, getting d′.
14 Return d′.
The following variations can be applied for further im-
provements:
1. If the dimension of V is larger than 1, there might
be different choices of witness vectors. Choosing dif-
ferent versions in line 2 might lead to different de-
nominator bounds of W, say, d ,...,d . Then taking
1 k
3.4 A Denominator Bounding Algorithm d := gcd(d ,...,d ) leads to a sharper denominator
1 k
We now combine Theorems 2 and 3 to an algorithm for boundfor (1) w.r.t. W.
computing a denominator bound with respect to an arbi-
2. Choosing different inner vectors in line 10 might lead
trarygivenW insituationswherethesetheoremsareappli-
todifferentsetsR− towrite(4)andhencegivesriseto
cable.
differentdenominatorboundsin(5). Takingthegcdof
Definition 3. Let p,p′ be corner points of S and W thesedenominator boundsproducesa refinedversion.
some submodule of r.
Z We remark that the coefficients as with s ∈ S are of-
1. The point p is called useless for W if there is an edge ten available in factorized form. Then also the denomina-
(p,s) in the convex hull of S\{p} with p−s∈W. tor bounds are obtained in factorized form, and the gcd-
computations reduce to comparisons of these factors and
2. The pair (p,p′) is called opposite if there is a vector
bookkeepingof their multiplicities.
v such that (s−p)·v≥0 and (p′−s)·v≥0 for all
s∈S. Such a v is called a witnessvector for the pair
4. ACOMBINEDDENOMINATORBOUND
(p,p′).
As mentioned earlier, when setting W = {0}, one is able
3. The pair (p,p′) is called useful for W if it is opposite
to derivean aperiodic denominatorbound for equation (1).
and neither p nor p′ is useless for W. In this particular case, for each corner point p there is an
othercornerpointp′suchthat(p,p′)isusefulforW. Hence
Thedefinitionofausefulpairismadeinsuchawaythat
applying Algorithm 1 for any useful pair leads to an aperi-
whenachangeofvariablesasdescribedinSection3.1isap-
odicdenominatorbound. Inparticular,runningthroughall
pliedwhichmapsawitnessvectorofthepairtothefirstaxis,
corner points and taking the gcd for all these candidates
thenthesetsAandB fromTheorem2aresuchthatp∈A,
leads to a rather sharp aperiodic denominator bound for
p′ ∈ B (because of the oppositeness), no two elements of
equation (1) which coincides with the output given in our
A agree in the first r −dimW coordinates (because p is
previousinvestigation [11].
not useless), and the same is true for B (because p′ is not Intheotherextreme,whensettingW = r,adenomina-
useless). Z
torboundfor(1)w.r.t.W wouldleadtoacompletedenom-
Whetherapair (p,p′)∈S2 is usefulor not can befound
inatorboundforequation(1). However,inthiscase,wewill
out by making an ansatz for the coefficients of a witness fail to find a useful pair (p,p′), and our Algorithm 1 is not
vectorandsolvingthesystemoflinearinequalitiesfromthe
applicable.
definition. The pair is useful if and only if this system is
Our goal is to find a simultaneous denominator bound
solvable,andinthiscase,anysolutiongivesrisetoawitness
with respect to all W to which Algorithm 1 is applicable,
vector.
i.e., for all W from theset
If for a given submodule W we have found a useful pair,
then we can computeadenominator boundwith respect to U :={W submoduleof r |∃ (p,p′) useful for W}.
W by thefollowing algorithm. Z
In general, this is an infinite set. But we can make use of
Algorithm 1. Input: An equation of the form (1), a theobservations madeafter Example 2. Using Lemma2, it
submodule W of r, a useful pair (p,p′) of S for W. Out- turnsoutthatinsteadofloopingthroughalltheseinfinitely
put: A denominaZtor bound for (1) with respect to W. manymodulesW,itissufficienttoconsiderthoseW which
appear as spread of some factor in thecoefficient of ap.
1 Set t:=r−dimW. This argument even works for all W in thelarger set
2 Choose v ,v ,...,v ∈ r such that v is a witness
1 2 t 1
3 vector for (p,p′)Zand r =V ⊕W where O:={W submoduleof r |
4 V is the module generaZted by these vectors. ∃ (p,p′) opposiZte with p not useless for W},
5 Perform a change of variables as described in
6 Section 3.1 such that v becomes the ith unit but since the W ∈ O\U do not satisfy the conditions of
i
7 vector in r, W becomes {0}t× r−t. Theorem 2, we can only obtain partial information about
8 Determine A,B Zas in Theorem 2. Z their denominatorbounds.
9 Compute s∈ ∪{−∞} as defined in Theorem 2. Wepropose thefollowing algorithm.
N
Algorithm 2. Input: Anequationoftheform (1). Out- some part of the denominator can be given up to possible
put: AfinitesetofirreduciblepolynomialsP ={p ,...,p }, shifts and multiplicities, and a big part of our denominator
1 k
andanonzerod∈ [n]suchthatforeverysolutiony= p ∈ boundcan begiven explicitly by d.
K q
(n) of (1) and every irreducible factor u of q with multi- The following improvementscan beutilized.
K
plicity m exactly one of the following holds:
1. Spread(u)∈U and um |d, 1. As preprocessing step, one should compute an aperi-
2. Spread(u)∈O\U and ∃ s∈ r,p∈P :Nsu=p, odic denominator bound for the equation (1) as de-
3. Spread(u)∈/ O. Z scribed above. What remains is to recover the peri-
odic factors. As a consequence, one can neglect all
1 d:=1
irreducible factors u which are aperiodic and one can
2 P :={}
apply Theorem 3 where all aperiodic factors are re-
3 C :={p∈S :p is a corner point of S} moved from thepolynomials a′.
4 forall q∈C do p
5 forall u|aq irreducible do 2. ChoosingdifferentusefulpairsforamoduleW inline9
6 W :=Spread(u) mightleadtodifferentchoicesofdenominatorbounds,
7 if W ∈U then andtakingtheirgcdgivesrisetosharperdenominator
8 Compute a denominator bound d w.r.t. W boundsof (1) w.r.t. W.
0
9 using an arbitrary useful pair for W.
10 d:=lcm(d,d0) 5. DISCUSSION
11 else if W ∈O then
12 P :=P ∪{u} TypicallythesetO willcontainallthesubmodulesofZr.
OnlywhentheconvexhullofShappenstohavetwoparallel
13 return (P,d)
edges on opposite sides, as is the case in Example 2, then
Theorem 4. The polynomial d computed by Algorithm 2 modules W parallel to this edge do not belong to O. The
setU willnevercontainallthesubmodulesof r. Precisely
is a denominator bound with respect to any finite union of
Z
thosemodulesW whichareparalleltoanedgeoftheconvex
modules in U.
hull of S do not belong to U. Since the convex hull of S
Proof. Let W be in U and (p,p′) be a useful pair with
contains only finitelymany edges, U will in some sense still
respect toW. Let y=p/q bea solution of (1) and u bean
contain almost all thesubmodulesof r.
irreducible factor of q with multiplicity m and Spread(u)⊆ Z
Depending on the origin of the equation, it may be that
W. Wehaveto show that um |d.
there is some freedom in the structure set S. For example,
Sincepisnotuseless,Lemma2impliesthatthereissome
by multivariate guessing [10] or by creative telescoping [16,
i∈Zr withNiu|ap. Thisfactorisgoingtobeinvestigated 9, 14] one can systematically search for equations with a
in someiteration oftheloop startingin line5. Thepolyno-
prescribed structure set. In such situations, one can try
mial d computed in this iteration is a denominator bound
0 to search for an equation with a structure set for which U
with respect to Spread(Niu) = Spread(u) and hence a for-
and O cover as many spaces as possible.
teriori a denominator bound with respect to W. It follows
Iftwoequationswithdifferentstructuresetsareavailable,
that um |d |d.
0 it may bepossible to combinethetwodenominator bounds
This proves the theorem when W itself is in U. If W is
obtained by Algorithm 2 to a denominator bound with re-
only a finite union of elements of U, the theorem follows
spect tothefull space r.
from hereby Lemma 1. Z
For W ∈ O\U, we can still apply Lemma 2 but Theo- Example 4. Consider the followingsystem of equations:
rem 2 is no longer applicable. This prevents us from com-
−(k+n+1)(2k+3n+1)y(n,k)
putingprecisedenominatorboundswithrespecttotheseW.
However,usingthesetP ={p ,...,p }returnedbytheal- +(k+n+4)(2k+3n+3)y(n,k+1)
1 k
gorithm wecanatleast saythatforeverydenominatorq of −(k+n+2)(2k+3n+4)y(n+1,k)
a solution y=p/q of (1) there exist m∈ and a finite set
+(k+n+5)(2k+3n+6)y(n+1,k+1)=0,
S′ ⊆ r such that N
Z (n2+n+1)(2k+3n+3)y(n,k+1)
d Nspm (6)
Y −(n2+5n+7)(2k+3n+4)y(n+1,k)
p∈P
s∈S′ −(n2+3n+3)(2k+3n+8)y(n+1,k+2)
is amultipleofevery divisorofq whose spread iscontained
in some finite union of modules in O. Appropriate choices +(n2+7n+13)(2k+3n+9)y(n+2,k+1).
S′ and m can be found for instance by making an ansatz.
Algorithm 2 applied to the first equation returns
Note also that the set P is usually smaller than the set of
all periodic factors that occur in the coefficients as of (1). d=(n+k+1)(n+k+2)(n+k+3)(3n+2k+1)
This phenomenon was demonstrated already in the second
as a denominator bound with respect to any W except 1
part of Example 2. (cid:0)0(cid:1)Z
Summarizing, some part of the denominator is out of and (cid:0)01(cid:1)Z. Applied to the second equation, it returns
reach, namely all those parts of the denominator w.r.t. the d=(n2+n+1)((n+1)2+(n+1)+1)(3n+2k+1)
modules from
{W submoduleof r | as a denominator bound with respect to any W except (cid:0)11(cid:1)Z
Z and 1 . Theleastcommonmultipleofthetwooutputs is
∀ (p,p′) opposite for W with p and p′ useless for W}, (cid:0)−1(cid:1)Z
a simultaneous denominator bound with respect to any W.
Indeed, the system has the solution equationsin ΠΣ-extensions.In Proc. SYNASC’04,
pages 269–282, 2004.
1 .
(n+k+1)(n+k+2)(n+k+3)(n2+n+1)((n+1)2+(n+1)+1)(3n+2k+1) [15] Mark van Hoeij. Rational solutions of linear difference
equations.In Proc. ISSAC’98,pages 120–123, 1998.
Thereisnohopeforanalgorithmwhichcomputesforany
[16] Doron Zeilberger. The method of creative telescoping.
given single equation a denominator bound with respect to
the full space r. This is because there are quations whose Journal of Symbolic Computation, 11:195–204, 1991.
Z
solutionspacecontainsrationalfunctionswithnofinitecom-
mondenominator. Forinstance,foreveryunivariatepolyno-
mialp,wehavethat1/p(n+k)isasolutionoftheequation
y(n+1,k)−y(n,k+1)=0.
It would beinteresting tocharacterize underwhich circum-
stances thishappens, and to havean algorithm which finds
a denominator boundwith respect to r in all othercases.
Z
6. REFERENCES
[1] S.A.AbramovandM.Barkatou.Rationalsolutionsof
first order linear differencesystems. In Proceedings of
ISSAC’98,pages 124–131, 1998.
[2] Sergei A.Abramov.On thesummation of rational
functions. Zh. vychisl. mat. Fiz, pages 1071–1075,
1971.
[3] Sergei A.Abramov.Problems in computeralgebra
that are connected with a search for polynomial
solutions of linear differentialand difference equations.
Moscow Univ. Comput. Math. Cybernet., 3:63–68,
1989.
[4] Sergei A.Abramov and K.Yu.Kvashenko.Fast
algorithms tosearch for therational solutions of linear
differential equationswith polynomial coefficients. In
Proc. ISSAC’91,pages 267–270, 1991.
[5] Moulay Barkatou. Rational solutions of matrix
differenceequations: The problem of equivalenceand
factorization. In Proc. ISSAC’99,pages 277–282, 1999.
[6] Alin Bostan, FredericChyzak,Thomas Cluzeau, and
BrunoSalvy.Low complexity algorithms for linear
recurrences. In Jean-Guillaume Dumas, editor, Proc.
ISSAC’06,pages 31–39, 2006.
[7] Manuel Bronstein. On solutions of linear ordinary
differenceequations in their coefficient field.Journal
of Symbolic Computation, 29:841–877, 2000.
[8] William Y.C.Chen,PeterPaule, andHusam L.Saad.
Converging to Gosper’s algorithm. Adv. in Appl.
Math., 41(3):351–364, 2008.
[9] Fr´ed´eric Chyzak.Anextension of Zeilberger’s fast
algorithm to general holonomic functions. Discrete
Mathematics, 217:115–134, 2000.
[10] Manuel Kauers. Guessing handbook.Technical Report
09-07, RISC-Linz,2009.
[11] Manuel Kauersand Carsten Schneider.Partial
denominatorboundsfor partial linear difference
equations. InProceedings of ISSAC’10,pages 211–218,
2010.
[12] Peter Paule. Greatest factorial factorization and
symbolic summation. Journal of Symbolic
Computation, 20:235–268, 1995.
[13] Marko Petkovˇsek.Hypergeometricsolutions of linear
recurrences with polynomial coefficients. Journal of
Symbolic Computation, 14(2-3):243–264, 1992.
[14] Carsten Schneider.A collection of denominator
boundstosolve parameterized linear difference