Order-Degree Curves for Hypergeometric Creative Telescoping

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1 Ode-Degee Cuves fo Hyegeometc Ceatve Telescong ABSTRACT Shaosh Chen Deatment of Mathematcs NCSU Ralegh, NC 7695, USA Ceatve telescong aled to a bvaate oe hyegeometc tem oduces lnea ecuence oeatos wth olynomal coeffcents, called telescoes We ovde bounds fo the degees of the olynomals aeang n these oeatos Ou bounds ae exessed as cuves n the (, d-lane whch assgn to evey ode a bound on the degee d of the telescoes These cuves ae hyebolas, whch eflect the henomenon that hghe ode telescoes tend to have lowe degee, and vce vesa Categoes and Subject Desctos I [Comutng Methodologes]: Symbolc and Algebac Manulaton Algothms Geneal Tems Algothms Keywods Symbolc Summaton, Ceatve Telescong, Degee Bounds INTRODUCTION We consde the oblem of fndng lnea ecuence equatons wth olynomal coeffcents satsfed by a gven defnte sngle sum ove a oe hyegeometc tem n two vaables Ths s one of the classcal oblems n symbolc summaton Zelbege [6] showed that such a ecuence always exsts, and oosed the algothm now named afte hm fo comutng one [5, 7] Also exlct bounds ae known fo the ode of the ecuence satsfed by a gven sum [4, 0, 4] Lttle s known howeve about the degees Suoted by the Natonal Scence Foundaton (NFS gant CCF-077 Suoted by the Austan Scence Fund (FWF gant Y464-N8 Pemsson to make dgtal o had coes of all o at of ths wok fo esonal o classoom use s ganted wthout fee ovded that coes ae not made o dstbuted fo oft o commecal advantage and that coes bea ths notce and the full ctaton on the fst age To coy othewse, to eublsh, to ost on seves o to edstbute to lsts, eques o secfc emsson and/o a fee Coyght 0XX ACM X-XXXXX-XX-X/XX/XX $000 Manuel Kaues RISC Johannes Kele Unvesty 4040 Lnz, Austa mkaues@scjkuat of the olynomals aeang n the ecuence These ae nvestgated n the esent ae Ideally, we would lke to be able to detemne fo a gven sum all the as (, d such that the sum satsfes a lnea ecuence of ode wth olynomal coeffcents of degee at most d Ths s a had queston whch we do not exect to have a smle answe The esults gven below can be vewed as answes to smlfed vaants of the oblem One smlfcaton s that we estct the attenton to the ecuences found by ceatve telescong [7], called telescoes n the symbolc summaton communty (see Secton below fo a defnton The second smlfcaton s that nstead of tyng to chaacteze all the as (, d, we confne ouselves wth suffcent condtons Ou man esults ae thus fomulas whch ovde bounds on the degee d of the olynomal coeffcents n a telescoe, deendng on ts ode The fomulas descbe cuves n the (, d-lane wth the oety that fo evey ntege ont (, d above the cuve, thee s a telescoe of ode wth olynomal coeffcents of degee at most d As the cuves ae hyebolas, they eflect the henomenon that hghe ode ecuence equatons may have lowe degee coeffcents Ths featue can be used to deve a comlexty estmate accodng to whch, at least n theoy, comutng the mnmum ode ecuence s moe exensve than comutng a ecuence wth slghtly hghe ode (but dastcally smalle olynomal coeffcents Ths henomenon s analogous to the stuaton n the dffeental case, whch was fst analyzed by Bostan et al [5] fo algebac functons, and ecently fo ntegals of hyeexonental tems by the authos [6] Ou analyss fo non-atonal oe hyegeometc nut (Secton 3 follows closely ou analyss fo the dffeental case [6] It tuns out that the summaton case consdeed hee s slghtly ease than the dffeental case n that t eques fewe cases to dstngush and n that the esultng degee estmaton fomula s much smle than ts dffeental analogue Fo atonal nut (Secton 4, we deve a degee estmaton fomula followng Le s algothm fo comutng telescoes of atonal functons [, 9] PROPER HYPERGEOMETRIC TERMS AND CREATIVE TELESCOPING Let K be a feld of chaactestc zeo and let K(n, k be the feld of atonal functons n n and k We wll be consdeng extenson felds E of K(n, k on whch two somohsms S n and S k ae defned whch commute wth each

2 othe, leave evey element of K fxed, and act on n and k va S n(n = n +, S k (n = n, S n(k = k, S k (k = k + A hyegeometc tem s an element h of such an extenson feld E wth S n(h/h K(n, k and S k (h/h K(n, k Poe hyegeometc tems ae hyegeometc tems whch can be wtten n the fom M h = x n y k Γ(a mn + a mk + a mγ(b mn b mk + b m Γ(u mn + u mk + u mγ(v mn v mk + v m, ( whee K[n, k], x, y K, M N s fxed, a m, a m, b m, b m, u m, u m, v m, v m ae nonnegatve nteges, a m, b m, u m, v m K and the exessons x n, y k, and Γ( efe to elements of E on whch S n and S k act as suggested by the notaton, eg, S n(γ(n k + = (n k+(n k+γ(n k +, S k (Γ(n k + = Γ(n k + n k Thoughout the ae the symbols, x, y, M, a m, a m, a m, wll be used wth the meanng they have n ( Fo fttng long fomulas nto the naow columns of ths layout, we also use the abbevatons A m := a mn + a mk + a m, B m := b mn + b mk + b m, U m := u mn + u mk + u m, V m := v mn + v mk + v m We follow the aadgm of ceatve telescong Fo a gven hyegeometc tem h as above, we want to detemne olynomals l 0,, l K[n] (fee of k, not all zeo, and a atonal functon C K(n, k (ossbly nvolvng k, ossbly zeo, such that l 0h + l S n(h + + l S n(h = S k (Ch Ch In ths case, the oeato L := l 0 + l S n + + l Sn K[n][S n] s called a telescoe fo h, and the atonal functon C K(n, k s called a cetfcate fo L (and h The numbe s called the ode of L, and d := max deg n l s called ts degee If h eesents an actual sequence f(n, k, then a ecuence fo the defnte sum n k=0 f(n, k can be obtaned fom such a a (L, C as exlaned n the lteatue on symbolc summaton [] We shall not embak on the techncal subtletes of ths coesondence hee but estct ouselves to analyzng of the set of all as (, d fo whch thee exsts a telescoe of ode and degee d The followng notaton wll be used Fo K[n, k] and m N, let m := ( + ( + ( + m wth the conventons 0 = and = Fo K[n, k], deg n and deg k denote the degee of wth esect to n o k, esectvely deg wthout any subsct denotes the total degee of Fo z R, let z + := max{0, z} Wth ths notaton, we have S n(h h S k (h h = x Sn( = y S k( A am m Bm bm U um m V vm m A a m m (V m v m v m U u m m (B m b m b m, 3 THE NON-RATIONAL CASE We consde n ths secton the case whee h cannot be slt nto h = qh 0 fo q K(n, k and anothe hyegeometc tem h 0 wth S k (h 0/h 0 = Infomally, ths means that we exclude tems h whee y = and evey Γ-tem nvolvng k can be cancelled aganst anothe one to some atonal functon Those tems ae teated seaately n Secton 4 below If h cannot be slt as ndcated, then also Ch cannot be slt n ths way, fo any atonal functon C K(n, k In atcula, we can then not have Ch K and theefoe we always have S k (Ch Ch 0 Ths mles that wheneve we have a a (L, C wth L(h = S k (Ch Ch, we can be sue that L s not the zeo oeato, and we need not woy about ths equement any futhe The analyss n the esent case s smla to that caed out by Aagodu and Zelbege [0], who used t fo devng a bound on the ode of L, and smla to ou analyss [6] of the dffeental case The man dea s to follow ste by ste the executon of Zelbege s algothm when aled to h Ths eventually leads to a lnea system of equatons wth coeffcents n K(n whch must have a soluton wheneve t s undedetemned The condton of havng moe vaables than equatons n ths lnea system s the souce of the estmate fo choces (, d that lead to a soluton 3 Zelbege s Algothm Recall the man stes of Zelbege s algothm: fo some choce of, t makes an ansatz L = l 0+l S n+ +l S n wth undetemned coeffcents l 0,, l, and then calls Gose s algothm on L(h Gose s algothm [7] oceeds by wtng S k (L(h L(h = S k(p P Q S k (R fo some olynomals P, Q, R such that gcd(q, S k(r = fo all N It tuns out that the undetemned coeffcents l 0,, l aea lnealy n P and not at all n Q o R Next, the algothm seaches fo a olynomal soluton Y of the Gose equaton P = Q S k (Y R Y by makng an ansatz Y = y 0 + y k + y k + + y sk s fo some sutably chosen degee s, substtutng the ansatz nto the equaton, and comang owes of k on both sdes Ths leads to a lnea system n the vaables l 0,, l, y 0,, y s wth coeffcents n K(n Any soluton of ths system gves se to a telescoe L wth the coesondng cetfcate C = RY/P If no soluton exsts, the ocedue s eeated wth a geate value of Fo a hyegeometc tem h and an oeato L = l 0 + l S n + + l S n, we have L(h = = l x S n( l x Sn( Um um Vm vm A am m Bm bm U um m P,m h V vm m ( = l x Sn( P,m h

3 whee x n y k M Γ(A mγ(b m Γ(U m + u mγ(v m + v m, P,m = A am m Bm bm (U m + u m ( um (V m + v m ( vm We can wte whee P = Q = y R = S k (L(h L(h = S k(p P l x Sn( Q S k (R, P,m, A a m m (V m + v m v m v m, (U m + u m u m u m B b m m Deendng on the actual values of the coeffcents aeang n h, ths decomoston may o may not satsfy the equement gcd(q, Sk(R = fo all N But even f t does not, t only means that we may ovelook some solutons, but evey soluton we fnd stll gves se to a coect telescoe and cetfcate Snce we ae nteested only n boundng the sze of the telescoes of h, t s suffcent to study unde whch ccumstances the Gose equaton P = Q S k (Y R Y wth the above choce of P, Q, R has a soluton 3 Countng Vaables and Equatons Aagodu and Zelbege [0] oceed fom hee by analyzng the lnea system ove K(n esultng fom the Gose equaton fo a sutable choce of the degee of Y They deve a bound on but gve no nfomaton on the degee d Geneal bounds fo the degees of solutons of lnea systems wth olynomal coeffcents could be aled, but they tun out to oveshoot qute much In atcula, t seems dffcult to catue the henomenon that nceasng may allow fo deceasng d usng such geneal bounds We oceed dffeently Instead of a coeffcent comason wth esect to owes of k leadng to a lnea system ove K(n, we consde a coeffcent comason wth esect to owes of n and k leadng to a lnea system ove K Ths eques us to make a choce not only fo the degee of Y n k but also fo the degee of Y n n as well as fo the degees of the l ( = 0,, n n Fo exessng the numbe of vaables and equatons n ths system, t s helful to adot the followng defnton Defnton Fo a oe hyegeometc tem h as n (, let δ = deg, { M M } ϑ = max (a m + b m, (u m + v m, λ = M (u m + v m, µ = M (a m + b m u m v m, { M M } ν = max (a m + v m, (u m + b m Note that these aametes ae nteges whch only deend on h but not on o d Excet fo µ, they ae all nonnegatve Note also that we have λ+µ 0 and ϑ = λ+µ + µ Lemma Let d := deg n l ( = 0,, Then deg P δ + λ + Futhemoe, deg k P δ + ϑ Poof It suffces to obseve that max (d + µ deg P,m a m + b m + ( u m + ( v m fo all m =,, M and all = 0,, Fo the degee wth esect to k, obseve also that deg k l = 0 fo all We have some feedom n choosng the d The choce nfluences the numbe of vaables n the ansatz d L = l,jn j Sn j=0 as well as the numbe of equatons We efe to have many vaables and few equatons Fo a fxed taget degee d, the maxmum ossble numbe of vaables s (d + ( + by choosng d 0 = d = = d = d But ths choce also leads to many equatons A bette balance between numbe of vaables and numbe of equatons s obtaned by loweng some of the d wth ndces close to zeo (f µ s negatve o wth ndces close to (f µ s ostve Secfcally, we choose { (ν + + µ f µ 0 d := d (ν + µ f µ < 0 See [6, Ex, Ex 55 and the emaks afte Thm 4] fo a detaled motvaton of the coesondng choce n the dffeental case The suot of the ansatz fo L looks as n the followng dagam, whee evey tem n j Sn s eesented by a bullet at oston (, j: j=d j=0 { }} { } = ν µ } Wth ths choce fo the degees d, the numbe of vaables n the ansatz fo L s (d + = (d + ( + µ ν(ν +, ovded that d µ ν The numbe of esultng equatons s as follows µ

4 Lemma 3 If the d ae chosen as above, then P contans at most ( ( δ + ϑ + δ + d + ϑ µ ν + tems n k j Poof If µ 0, we have d + µ = d (ν + + µ + µ d ν µ + µ fo all = 0,, Lkewse, when µ < 0, we have d + µ = d (ν + µ + µ d ν µ fo all = 0,, Togethe wth Lemma, t follows that deg P δ + (λ + µ + + d ν µ = δ + ϑ + d ν µ egadless of the sgn of µ We also have deg k P δ + ϑ fom Lemma Fo the numbe of tems n k j n P we have ( + deg P = (deg k P + ( deg P + deg k P deg k P Pluggng the estmates fo deg P and deg k P nto the ght hand sde gves the exesson clamed n the Lemma The suot of P has a taezodal shae whch s detemned by the total degee and the degee wth esect to k: deg P } {{ } deg k P The next ste s to choose the degees fo Y n n and k Ths s done n such a way that Q S k (Y R Y only contans tems whch ae aleady exected to occu n P, so that no addtonal equatons wll aea Lemma 4 Let the d be chosen as befoe and suose that Y K[n, k] s such that deg Y deg P ν and deg k Y deg k P ν Then P (Q S k (Y R Y contans at most tems n k j ( δ + ϑ + ( δ + d + ϑ µ ν + Poof As fo Lemma 3, usng also max{deg Q, deg R} = max{deg k Q, deg k R} = ν Lemma 4 suggests the ansatz Y = s s j=0 y,jk n j wth s = deg k P ν and s = deg P ν, whch ovdes us wth ( ( δ + ϑ + ν δ + d + ϑ µ ν + ν vaables We ae now eady to fomulate the man esult of ths secton Note that the nequalty fo d s a consdeably smle fomula than the coesondng esult n the dffeental case (Thm 4 n [6] Theoem 5 Let h be a oe hyegeometc tem whch cannot be wtten h = qh 0 fo some q K(n, k and a hyegeometc tem h 0 wth S k (h 0/h 0 = Let δ, λ, µ, ν be as n Defnton, let ν and d > ( ϑν + ν( δ + µ + 3 ( + µ ν ν + Then thee exsts a telescoe L fo h of ode and degee d Poof A suffcent condton fo the exstence of a telescoe of ode and degee d s that fo some atcula ansatz, the equaton P = Q S k (Y R Y has a nontval soluton A suffcent condton fo the exstence of a soluton s that the lnea system esultng fom coeffcent comason has moe vaables than equatons Fo all d n queston, we have d > ϑν µ ν Theefoe, wth the ansatz descbed above, we have (d + ( + µ ν(ν + vaables l,j n P, ( ( δ + ϑ + ν δ + d + ϑ µ ν + ν vaables y,j n Y, and ( ( δ + ϑ + δ + d + ϑ µ ν + equatons Solvng the nequalty (d + ( + µ ν(ν + ( ( δ + ϑ + ν δ + d + ϑ µ ν + ν > + ( δ + ϑ + ( δ + d + ϑ µ ν + unde the assumton ν fo d gves the clamed degee estmate 33 Examles and Consequences Examle 6 Fo h = (n + k Γ(n + 3k + Γ(n k we have δ =, ϑ =, µ = 0, ν = 4 Theoem 5 edcts a telescoe of ode and degee d wheneve 4 and d > The left fgue below shows the cuve defned by the ght hand sde (black togethe wth the egon of all onts (, d fo whch we found telescoes of h wth ode and degee d by dect calculaton (gay In ths examle, the estmate oveshoots by vey lttle only The coesondng ctue fo h = Γ(n + kγ(n k + Γ(n kγ(n + k s shown below on the ght Hee, δ = 0, ϑ = 3, µ = 0, ν = 3 and Theoem 5 edcts a telescoe of ode and degee d wheneve 3 and d > 8 In ths examle, the estmate s less tght

5 40 30 d d Theefoe, n ode to comute a telescoe and ts cetfcate most effcently, we should mnmze the cost functon C(, d := ( (ϑ + + δ + 3( δ + ϑ + d ( µ + ν The onts (, d n the oton of the gay egon whch s below the black cuve eesent telescoes whee the coesondng lnea system esultng fom the ansatz consdeed n ou oof s ovedetemned but, fo some stange eason, nevetheless nontvally solvable The small otons of whte sace whch le above the cuves ae not n contadcton wth ou theoem because they do not contan any onts wth ntege coodnates (The theoem says that evey ont (, d Z above the cuve belongs to the gay egon Theoem 5 sulements the bound gven n [0] on the ode of telescoes fo a hyegeometc tem by an estmate fo the degee that these oeatos may have In addton, t ovdes lowe degee bounds fo hghe odes and admts a bound on the least ossble degee fo a telescoe Coollay 7 Wth the notaton of Theoem 5, h admts a telescoe of ode = ν and degee d = ν(δ + νϑ + µ ν µ as well as a telescoe of ode and degee d = ϑν = ν( + δ + (ν (ϑ µ Poof Immedate by checkng that the two choces fo and d satsfy the condtons stated n Theoem 5 An accuate edcton fo the degees of the telescoes can also be used fo movng the effcency of ceatve telescong algothms Although most mlementatons today comute the telescoe wth mnmum ode, t may be less costly to comute a telescoe of slghtly hghe ode If we know n advance the degees d of the telescoes fo evey ode, we can select befoe the comutaton the ode whch mnmzes the comutatonal cost Of couse, the cost deends on the algothm whch s used It s not necessay (and not advsable to follow the stes n the devaton of Theoem 5 and do a coeffcent comason ove K Instead, one should follow the common actce [8] of comang coeffcents only wth esect to owes of k and solve a lnea system ove K(n Fo nonmnmal choces of, ths system wll have a nullsace of dmenson geate than one, of whch we need not comute a comlete bass, but only a sngle vecto wth comonents of low degee Thee ae algothms known fo comutng such a vecto usng O(m 3 t feld oeatons when the system has at most m vaables and equatons and the soluton has degee at most t [3, 3, 5] In the stuaton at hand, we have m = (++(δ+ϑ+ vaables and a soluton of degee t = δ+ϑ+d ( µ +ν+ Accodng to the followng theoem, fo asymtotcally lage nut t s sgnfcantly bette to choose slghtly lage than the mnmal ossble value Theoem 8 Let h and λ, µ, ν be as n Theoem 5, τ max{ϑ, ν}, and suose that κ R s a constant such that degee t solutons of a lnea system wth m vaables and at most m equatons ove K(n can be comuted wth κm 3 t oeatons n K Then: A telescoe of ode = τ along wth a coesondng cetfcate can be comuted usng oeatons n K κτ 9 + (7 µ κτ 8 + O(τ 7 If α > s some constant and s chosen such that = ατ + O(, then a telescoe of ode and a coesondng cetfcate can be comuted usng oeatons n K α 5 α κτ 8 + O(τ 7 In atcula, a telescoe fo h and a coesondng cetfcate can be comuted n olynomal tme Poof Accodng to Theoem 5, fo evey τ thee exsts a telescoe of ode and degee d fo any d > f( := (τ + O(τ τ By assumton, such a telescoe can be comuted usng no moe than C(, d = κ ( (τ + + δ + 3( δ + τ + d ( µ + τ + oeatons n K The clam now follows fom the asymtotc exansons of C(τ, f(τ+ and C(ατ, f(ατ+ fo τ, esectvely The leadng coeffcent n at s mnmzed fo α = 5/4 Ths suggests that when ϑ and ν ae lage and aoxmately equal, the ode of the cheaest telescoe s about 0% lage than the mnmal exected ode 4 THE RATIONAL CASE We now tun to the case whee h can be wtten as h = qh 0 fo some hyegeometc tem h 0 wth S k (h 0/h 0 = By the followng tansfomaton, we may assume wthout loss of genealty h 0 = Lemma 9 Let h be a hyegeometc tem and suose that h = qh 0 fo some q K(n, k and a hyegeometc tem h 0 wth S k (h 0/h 0 = Let a, b K[n, k] be such that S n(h 0/h 0 = a/b Let L be a telescoe fo q of ode and degee d Then thee exsts a telescoe fo h of ode and degee at most d + max{deg n a, deg n b}

6 Poof Wte L = l 0 + l S n + + l Sn and let C K(n, k be a cetfcate fo L and q, so L(q = S k (Cq Cq Fo = 0,,, let b ( b l := l a Sn a S n and L := l 0 + l S n + + l S n Then ( b a L(qh 0 = L(qh 0 = (S k (Cq Cqh 0 = S k (Cqh 0 Cqh 0 Because of ( a S k b = S k(s n(h 0 S k (h 0 = Sn(S k(h 0 S k (h 0 = Sn(h0 h 0 = a b, the oeato L s fee of k Thus, afte cleang denomnatos, L s a telescoe fo h wth coeffcents of degee at most d + max{deg n a, deg n b} Fom now on, we assume that h s at the same tme a oe hyegeometc tem and a atonal functon, o equvalently, that h s a atonal functon whose denomnato factos nto ntege-lnea factos Le [9] gves a ecse descton of the stuctue of telescoes n ths case, and he ooses an algothm dffeent fom Zelbege s fo comutng them Ou degee estmate s deved followng the stes of hs algothm, so we stat by befly summazng the man stes of Le s aoach 4 Le s Algothm Gven a atonal oe hyegeometc tem h, Le s algothm comutes a telescoe L fo h as follows Comute g K(n, k and olynomals, q K[n, k] wth gcd(q, S k(q = fo all Z \ {0} such that h = S k (g g + q Then an oeato L s a telescoe fo h f and only f L s a telescoe fo Abamov [] and Paule [] q exlan how to comute such a decomoston Comute a olynomal u K[n], oeatos V,, V s n K[n][S n], and atonal functons f,, f s of the fom f = (a n + k + a e ( =,, s such that q = s V (f u Such data always exsts accodng to Lemma 5 n [9] n combnaton wth the assumton gcd(q, Sk(q = ( Z \ {0} It can be futhe assumed that the f ae chosen such that > 0, e > 0, gcd(a, = fo all, and ( a aj n + a j fo all j wth e = e j ( a a j a Z j 3 Fo =,, s, comute an oeato L K(n[S n] such that S a n s a ght dvso of L ( V It follows u fom Le s Lemma 4 that the oeatos L wth ths oety ae ecsely the telescoes of the atonal functons V (f 4 Comute a common left multle L K[n][S n] of the oeatos L,, L s Then L s a telescoe fo h The man at of the comutatonal wok haens n the last two stes It theefoe aeas sensble to assume n the followng degee analyss that we aleady know the data u, V,, V s, f,, f s comuted n ste, and to exess the degee bounds n tems of the degees and coeffcents athe than n tems of the degees of numeato and denomnato of h, say 4 Countng Vaables and Equatons Also n the esent case, the degee estmate s obtaned by balancng the numbe of vaables and equatons of a cetan lnea system ove K The lnea system we consde ognates fom a atcula way of executng stes 3 and 4 of the algothm outlned above Theoem 0 Let u K[n] and let V,, V s K[n][S n] be oeatos of degee δ ( =, s Let f = (a n + k + a e fo some a K, a, Z wth > 0 and gcd(a, =, e > 0, suose ( a aj n + a j ( a a j a Z j fo all j wth e = e j Let h = s u V(f Then fo evey s a and evey d > + + s δ s + deg n u thee exsts a telescoe L fo h of ode and degee d Poof Accodng to Le s algothm, t suffces to fnd some L K[n][S n] and oeatos R K(n[S n] wth the oety that L( u V = R(Sa n fo all Denote by ρ the ode of V Wtng d := d deg n u, we make an ansatz L = Lu wth L = d j=0 l,jn j S n so that L has degee d and L u V = LV ( =,, s It thus emans to constuct oeatos R K[n][S n] wth LV = R (S a n Snce LṼ has ode + ρ and degee d + δ, we consde ansatzes fo the R of ode + ρ and degee d + δ, esectvely, because S a n and degee 0 Then we have altogethe ( + ( d + + has ode s ( + ρ + ( d + δ + vaables n L and the R, and comang coeffcents wth esect to n and S n n all the equed denttes LV = R (S a n leads to a lnea system wth s ( + ρ + ( d + δ + equatons Ths system wll have a nontval soluton wheneve the numbe of vaables exceeds the numbe of equa-

7 tons Unde the assumton s a, the nequalty ( + ( d s + + ( + ρ + ( d + δ + > s ( + ρ + ( d + δ + s equvalent to Ths comletes the oof d > + s + s a a δ 43 Examles and Consequences Examle h = The atonal functon (n 3k(3n k (n + k + (n + k + (n + k + (3n + k + can be wtten n the fom h = 4 u V(f whee u = (n n(n + 3(n (3n + (5n +, the f ae such that a = a = a 3 =, a 4 =, and the V ae such that δ = = δ 4 = 6 Theefoe, Theoem 0 edcts a telescoe of ode and degee d wheneve 5 and d > Ths cuve togethe wth the egon of all onts (, d fo whch a telescoe of ode and degee d exsts s shown n the left fgue below The coesondng ctue fo the atonal functon d h = (n k + (n 3k + 5 (n + k + 3(n + k + 5(n + k + (n + k + s shown n the fgue below on the ght Ths nut can be wtten n the fom h = S k (g g + 4 V (f u wth g = 0(n+3 (n+4(n+k+7, u = (n 4 (n+9(n+k+3(n+k+4 (3n+ (n 4 (n (n + 5(n + 9, the f such that a = a = a 3 =, a 4 =, and the V such that δ = 8, δ = δ 3 = δ 4 = 7 Accodng to Theoem 0, we theefoe exect a telescoe fo h of ode and degee d wheneve 5 and d > In ths examle, the estmate s not as tght as n the evous one d Agan, t s an easy matte to secalze the geneal degee bound to a degee estmate fo a low ode telescoe, o to an ode estmate fo a low degee telescoe Coollay Wth the notaton of Theoem 0, h admts a telescoe of ode = s a and degee d = deg n u + s (δ a as well as a telescoe of ode = s aδ and degee d = deg n u Poof Clea by checkng that the oosed choces fo and d ae consstent wth the bounds n Theoem 0 Also lke n the non-atonal case, the bounds fo the degees of the telescoes can be used fo devng bounds on the comutatonal cost fo comutng them In the esent stuaton, let us assume fo smlcty that the cost of stes and of Le s algothm s neglgble, o equvalently, that the nut h s of the fom s u V(f wth V K[n][Sn] We shall analyze the algothm whch caes out stes 3 and 4 of Secton 4 n one stoke by makng an ansatz ove K(n fo an oeato L = l 0 + l S n + + l Sn, comutng the ght emndes of L u V wth esect to Sa n and equatng the coeffcents to zeo We assume, as befoe, that the esultng lnea system s solved usng an algothm whose untme s lnea n the outut degee and cubc n the matx sze Then the algothm eques O( 3 d oeatons n K Theoem 3 Let u K[n], V,, V s K[n][S n], and f,, f s K(n, k be as n Theoem 0 and consde h = s u V(f Suose that κ R s a constant such that degee t solutons of a lnea system wth m vaables and at most m equatons ove K(n can be comuted wth κm 3 t oeatons n K Assume δ = = δ s =: δ > 0 and a = a = = a s =: a ae fxed Then: A telescoe of ode = a s can be comuted usng oeatons n K a 4 (δ κ s 4 + O(s 3 If α > s some constant and s chosen such that = αa s + O( then a telescoe of ode can be comuted usng α 3 α a 3 (δ + (α deg n uκ s 3 + O(s oeatons n K In atcula, a telescoe fo h can be comuted n olynomal tme Poof Accodng to the fst estmate stated n Theoem 0, fo evey a s thee exsts a telescoe of ode and degee d fo any d > f( := sa δ + sa + deg n u By assumton, such a telescoe can be comuted usng no moe than C(, d := κ 3 d oeatons n K The clam now follows fom the asymtotc exansons of C(a s, f(a s + and C(αa s, f(αa s + fo s, esectvely

8 When deg n u = 0, the leadng coeffcent n at s mnmzed fo α = 3/ Ths suggests that when s s lage and all the δ, and ae aoxmately equal, the ode of the cheaest oeato exceeds the mnmal exected ode by aound 50% It must not be concluded fom a lteal comason of the exonents n Theoems 5 and 0 that Le s algothm s faste than Zelbege s, because τ n Theoem 5 and s n Theoem 0 measue the sze of the nut dffeently Nevetheless, t s lausble to exect that Le s algothm s faste, because t fnds the telescoes wthout also comutng a (otentally bg coesondng cetfcate Ou man ont hee s not a comason of the two aoaches, but athe the obsevaton that both of them admt a degee analyss whch fts to the geneal aadgm that nceasng the ode can cause a degee do whch s sgnfcant enough to leave a tace n the comutatonal comlexty It can also be agued that the stuatons consdeed n Theoems 8 and 3 ae chosen somewhat abtaly (ϑ and ν gowng whle µ emans fxed; es s gowng whle all the δ and eman fxed Indeed, t would be wong to take these theoems as an advce whch telescoes ae most easly comuted fo a atcula nut at hand Instead, n ode to seed u an actual mlementaton, one should let the ogam calculate the otmal choce fo fom the degee estmates gven Theoems 5 and 0 wth the atcula aametes of the nut Unfotunately, we ae not able to llustate the seedu obtaned n ths way by an actual untme comason fo a concete examle, because fo examles whch can be handled on cuently avalable hadwae, the comutatonal cost tuns out to be mnmzed fo the least ode oeato But aleady fo examles whch ae only slghtly beyond the caacty of cuent machnes, the degee edctons n Theoems 5 and 0 ndcate that comutng the telescoe of ode one moe than mnmal wll stat to gve an advantage We theefoe exect that the esults esented n ths ae wll contbute to the movement of ceatve telescong mlementatons n the vey nea futue [7] Wllam Gose Decson ocedue fo ndefnte hyegeometc summaton Poceedngs of the Natonal Academy of Scences of the Unted States of Ameca, 75:40 4, 978 [8] Chstoh Koutschan A fast aoach to ceatve telescong Mathematcs n Comute Scence, 4( 3:59 66, 00 [9] Ha Q Le A dect algothm to constuct the mnmal Z-as fo atonal functons Advances n Aled Mathematcs, 30( :37 59, 003 [0] Mohamud Mohammed and Doon Zelbege Sha ue bounds fo the odes of the ecuences outut by the Zelbege and q-zelbege algothms Jounal of Symbolc Comutaton, 39(:0 07, 005 [] Pete Paule Geatest factoal factozaton and symbolc summaton Jounal of Symbolc Comutaton, 0:35 68, 995 [] Mako Petkovšek, Hebet Wlf, and Doon Zelbege A = B AK Petes, Ltd, 997 [3] Ane Stojohann and Glles Vllad Comutng the ank and a small nullsace bass of a olynomal matx In Poceedngs of ISSAC 05, ages , 005 [4] Hebet S Wlf and Doon Zelbege An algothmc oof theoy fo hyegeometc (odnay and q multsum/ntegal denttes Inventones Mathematcae, 08: , 99 [5] Doon Zelbege A fast algothm fo ovng temnatng hyegeometc denttes Dscete Mathematcs, 80:07, 990 [6] Doon Zelbege A holonomc systems aoach to secal functon denttes Jounal of Comutatonal and Aled Mathematcs, 3:3 368, 990 [7] Doon Zelbege The method of ceatve telescong Jounal of Symbolc Comutaton, :95 04, 99 5 REFERENCES [] Sege A Abamov Indefnte sums of atonal functons In Poceedngs of ISSAC 95, ages , 995 [] Sege A Abamov and Ha Q Le A cteon fo the alcablty of Zelbege s algothm to atonal functons Dscete Mathematcs, 59( 3: 7, 00 [3] Benhad Beckemann and Geoge Labahn A unfom aoach fo the fast comutaton of matx-tye Padé aoxmants SIAM Jounal on Matx Analyss and Alcatons, 5(3:804 83, 994 [4] Aln Bostan, Shaosh Chen, Fédéc Chyzak, and Zmng L Comlexty of ceatve telescong fo bvaate atonal functons In Poceedngs of ISSAC 0, ages 03 0, 00 [5] Aln Bostan, Fédéc Chyzak, Buno Salvy, Gégoe Lecef, and Éc Schost Dffeental equatons fo algebac functons In Poceedngs of ISSAC 07, ages 5 3, 007 [6] Shaosh Chen and Manuel Kaues Tadng ode fo degee n ceatve telescong Techncal Reot , AXv, 0