The complete Generating Function for Gessel Walks is Algebraic

THE COMPLETE GENERA TING FUNCTION F O R GESSEL W ALKS IS ALGEBRAIC ALIN BOST AN AND MANUEL KA UERS Abstract. Gessel wa lks are l attice wa lks in the quarter plane N 2 which st art at the or i- gin (0 , 0) ∈ N 2 and consist only of s teps ch osen fr om the set {← , ւ , ր , →} . W e prov e that if g ( n ; i, j ) denotes the n um ber of Gessel w alks of l ength n whic h end at the p oint ( i, j ) ∈ N 2 , then the triv ariate generating series G ( t ; x, y ) = X n,i,j ≥ 0 g ( n ; i, j ) x i y j t n is an a lgebraic function. 1. Introduction The star ting questio n in la ttice path theory is the following: How man y wa ys are ther e to w alk from the origin thro ugh the lattice Z 2 to a sp ecified p oint ( i, j ) ∈ Z 2 , us ing a fixed num ber n of steps chosen from a giv en set S of admissible steps. The question is not hard to answ er. If w e write f ( n ; i, j ) for this num ber and define the ge nerating function F ( t ; x, y ) := ∞ X n =0  X i,j ∈ Z f ( n ; i, j ) x i y j  t n ∈ Q [ x, y , x − 1 , y − 1 ][[ t ]] then a simple calculation suffices to see that F ( t ; x, y ) is r ational, i.e., it agrees with the series expansion at t = 0 of a certain rationa l function P /Q ∈ Q ( t, x, y ). This is element ary and w ell- known. Matters are getting more interesting if restrictions are imp osed. F or ex a mple, the gener ating function F ( t ; x, y ) w ill typically no lo ng er b e ra tional if lattice paths are considered which, as befo re, start at the orig in, consist o f n steps, end at a given p oint ( i , j ), but which, as an a dditional requirement, never step out of the right half-plane. In was shown in [ 8 , Prop. 2] that no matter which set S of a dmissible steps is chosen, the co mplete g e nerating function F for such walks is algebraic , i.e., it satisfies P ( F, t, x, y ) = 0 for some po lynomial P ∈ Q [ T , t, x, y ]. If the walks a re no t restricted to a half-plane but to a qua r ter plane, say to the fir st quadr ant, then the generating function F might not even be algebraic . F or some step sets it is, for others it is not [ 6 , 23 ]. Among the s e ries which ar e not algebra ic, there ar e some which a re s till D-finite with r esp ect to t (i.e., they satisfy a linea r differ ential equation in t with p o lynomial co efficients in Q [ t, x, y ]), and others which ar e not even tha t [ 8 , 24 ]. Bousquet-M´ elou a nd Mishna [ 7 ] hav e sys tematically inv estigated all the walks in the quarter plane with step sets S ⊆ {← , տ , ↑ , ր , → , ց , ↓ , ւ} . After discarding trivia l cases and a pplying symmetries, they reduced the 256 different step sets to 79 inherently different cases to s tudy . They provided a unified wa y to pr ov e that 22 of those ar e D-finite, and gav e striking e vidence that 5 6 are not D-finite. Only a single s tep set sus tained their a ttacks, and this is the step set that we ar e considering her e . This critical step set is {← , ւ , ր , →} . The central o b ject of the present ar ticle are thus lattice walks in Z 2 which • star t at the origin (0 , 0), • consis t of n steps chosen from the step set { ← , ւ , ր , →} , and • never s tep o ut of the first quadra nt N 2 of Z 2 . 2000 Mathematics Subje ct Classific ation. Pr imary 05A15, 14N10, 33F10, 68W30; Secondary 33C05, 97N80. Key wor ds and phr a ses. Combinatorial enumeration, generating function, lattice walks, Gessel conjecture, alge- braic functions, computer algebra, automated guessing, fast algorithms. 1 2 ALIN BOST AN AND MANUEL KAUERS These w alks a r e also known a s Gessel walks . By g ( n ; i, j ), we deno te the num ber of Gessel walks of length n which end at the p oint ( i, j ) ∈ Z 2 . The complete genera ting function o f this seq ue nce is denoted by G ( t ; x, y ) = ∞ X n =0  X i,j ∈ Z g ( n ; i, j ) x i y j  t n . Since g ( n ; i, j ) = 0 if min ( i, j ) > n or max( i, j ) < 0, the inner sum is a p o lynomial in x and y for every fixed choice of n , and thus G ( t ; x, y ) lives in Q [ x, y ][[ t ]]. Gessel [unpublished] cons idered the sp ecial end p o int i = j = 0, i.e., Gess el walks returning to the or igin, so -called excursions . Their co unt ing sequence g ( n ; 0 , 0) starts a s 1 , 0 , 2 , 0 , 1 1 , 0 , 85 , 0 , 78 2 , 0 , 80 04 , 0 , 8 8044 , 0 , 1 0 2016 2 , 0 , . . . He observed empir ic a lly that these num bers a dmit a s imple hypergeo metric c lo sed form. His observ ation b ecame known as the Gessel c onje ctur e , and remained op en fo r several years. Only recently , it was shown to b e true: Theorem 1. [ 16 ] G ( t ; 0 , 0) = 3 F 2  5 / 6 1 / 2 1 5 / 3 2     16 t 2  = ∞ X n =0 (5 / 6) n (1 / 2) n (5 / 3) n (2) n (4 t ) 2 n . This res ult o bviously implies that G ( t ; 0 , 0) is D-finite. Less obvious a t this p oint, a nd a ctually ov erlo oked until now, is the fact that the p ower series G ( t ; 0 , 0) is even algebr ai c. B ecause of the alternative r e pr esentation 3 F 2  5 / 6 1 / 2 1 5 / 3 2     16 t 2  = 1 t 2  1 2 2 F 1  − 1 / 6 − 1 / 2 2 / 3     16 t 2  − 1 2  it was clear that algebr aicity could be decided by re fer ence to Sch w arz’s clas s ification [ 30 ] o f algebraic 2 F 1 ’s, but as no b o dy recognized that the parameters ( − 1 / 6 , − 1 / 2; 2 / 3) actually fit to Case I I I of Schw arz’s table, the rumor started to circulate that G ( t ; 0 , 0 ) is not algebr aic. In fact: Corollary 2 . G ( t ; 0 , 0) is algebr ai c. With Theore m 1 and sta nda rd softw are pack ages like gfun [ 29 , 21 ] at ha nd, discov ering and proving Cor. 2 is an easy co mputer algebra exercis e. Compared to a pro of by table- lo okup, the constructive pr o of given below has the adv a nt age that it applies similarly a ls o for fa milies of functions for which classificatio n results ar e not av ailable. Pr o of. The idea is to c ome up with a p o ly nomial P ( T , t ) in Q [ T , t ] a nd prov e that P admits the power ser ie s g ( t ) = P ∞ n =0 (5 / 6) n (1 / 2) n (5 / 3) n (2) n (16 t ) n as a r o ot. Using Thm. 1 , this implies that P ( T , t 2 ) is an annihilating p olynomia l for G ( t ; 0 , 0), so that the latter s e r ies is indeed a lgebraic. Such a poly nomial P can b e guesse d sta rting from the first, say , 1 00 terms, o f the s eries g ( t ), using for instance M aple ’s routine seriestoalg eq fr om the gfun pack age (see Sectio ns 2.1 and 3.1 fo r more details on auto ma ted g uessing). The explicit for m of P is g iven b elow. By the implicit function theo rem, that poly no mial P a dmits a r o ot r ( t ) ∈ Q [[ t ]] with r (0) = 1. Since P ( T , 0) = T − 1 ha s a single ro ot in C , the series r ( t ) is the unique ro ot of P in C [[ t ]]. Now, r ( t ) b eing alg ebraic, it is D-finite, and thus its c o efficients sa tisfy a recurr ence with p olyno mial co efficients. T o complete the pro of, it is then sufficient to type the following commands int o Maple . > with(g fun): > P:=(t, T) -> -1+4 8*t-5 76*t^ 2-256*t^3+(1-60*t+912*t^2-512*t^3)*T+(10*t -312*t ^2+62 4*t^3-512*t^4)*T^2+(45*t^2-504*t^3-576*t^4)*T^3+(117*t^3 -252*t ^4-28 8*t^5)*T^4+189*t^4*T^5+189*t^5*T^6+108*t^6*T^7+27*t^7*T^8: > gfun:- diffe qtore c(gfun:-algeqtodiffeq(P(t,r), r( t)), r(t), g(n)); This o utputs the first-order r ecurrence ( n + 2 )(3 n + 5) g n +1 − 4(6 n + 5 )(2 n + 1 ) g n = 0 , g 0 = 1 , satisfied b y the co efficients of r ( t ) = P ∞ n =0 g n t n . Its so lution is g n = (5 / 6) n (1 / 2) n (5 / 3) n (2) n 16 n , and there fore g ( t ) and r ( t ) coincide, and thus g ( t ) is a solution o f P , as was to be shown.  THE COMPLETE GENERA TING FUNCTION FOR GESSEL W ALKS IS ALGEBRAIC 3 The aim in the pre sent article is to lift the re s ult of Coro llary 2 to the complete generating function, wher e x and y are kept as parameters. W e a re go ing to show: Theorem 3. G ( t ; x, y ) is algebr a ic. This tw ofold gener alization of Thm. 1 is a surpris ing res ult. Un til now, it was not known whether G ( t ; x, y ) is even D-finite w ith r esp ect to t o r not, a nd bo th case s se e med equally plausible in view of known results ab out other step sets. T hm. 3 implies that G ( t ; x, y ) is D-finite with r e sp ect to ea ch of its v ariables, and in particula r that the sequence g ( n ; i, j ) is P-finite (i.e., it sa tisfies a line a r re c urrence with po lynomial co efficients in n ) for a ny choice of ( i, j ) ∈ N 2 . This s e ttles several conjectures made by Petko v ˇ sek and Wilf in [ 26 , § 2]. As noted in [ 26 ], even for simple v alues of ( i, j ) the s e q uence g ( n ; i, j ) is not hyperg eometric, unlike the exc ur sions s equence g (2 n ; 0 , 0). F or instance , the sequence g (2 n + 1; 1 , 0) satisfies a third order linear recurr ence, but it is not hypergeometr ic. Mo reov er, no clo sed formula seems to exis t for g ( n ; i, j ), for ar bitrary ( i, j ). All this indica tes that c o unting ge ne r al walks is muc h more difficult that just c o unting excur sions. Theorem 3 will be es tablished by obstinately using the approach based on automatic guessing and pr o of promoted in [ 5 ], and by making heavy use of co mputer algebra . In contrast to Corol- lary 2 , we mana ge in our pro of of Theorem 3 to avoid exhibiting a p olynomia l that has G ( t ; x, y ) as a ro o t. This is for tunate, since a po sterior i estimates show that the minimal p olynomial of G ( t ; x, y ) is huge, having a tota l size o f a bo ut 30Gb. Only annihilating p olyno mials of the sectio n ser ies G ( t ; x, 0) and G ( t ; 0 , y ) a re pro duced a nd manipulated during the computer-dr iven pro of of Theor em 3 . B ut even r estricted to those ones, our computations hav e led to expressions far to o la r ge to b e included into a printed publication; to o la rge even to b e pro ces sed efficiently by standar d computer a lgebra systems like Map le or Mathematica . T o get the computations completed, it was necessa ry to use careful implementations of so phisticated sp ecial purp ose alg orithms, and to run these on computers equipp ed with fast pro cesso rs and larg e memory capa cities. Thes e computations w ere p erformed using the computer algebra sys tem Mag ma [ 2 ]. O ur result is therefor e interesting not o nly b ecause of its combinatorial significance, but it is a lso no teworthy b ecause of the immense computationa l effor t that was deploy ed to establish it. 2. A Dr y R un: Kreweras w alks The computations which were neede d for pr oving Thm. 3 were p erfor med by means o f efficient sp ecial purp ose softw are running on fast hardware. It would no t b e easy to re do these calcula tions in, s ay , M aple o r Mathematic a on a standard computer. As a more easily repro ducible calculation, we will show in this section how to repr ov e the class ical res ult that the genera ting function o f Kreweras walks is algebr aic [ 19 , 1 4 , 6 ]. A slight v ar iation o f the very s a me reasoning, alb eit with int ermediate e xpressions far to o la rge to b e sp elled out here, is then used in the next section to establish Thm. 3 . Kreweras walks differ from Gessel walks only in their c hoice of admissible steps. They are thus defined a s lattice w alks in Z 2 which • star t at the origin (0 , 0), • consis t only of steps chosen from the step set {← , ↓ , ր} , and • never s tep o ut of the first quadra nt N 2 of Z 2 . If f ( n ; i, j ) denotes the num b er of Kreweras walks consisting o f n s teps and ending at the p oint ( i, j ) ∈ Z 2 , then it follows directly from its co mbinatorial definitio n that the s equence f ( n ; i, j ) satisfies the multiv ariate r ecurrence with constant co efficients (1) f ( n + 1; i, j ) = f ( n ; i + 1 , j ) + f ( n ; i, j + 1 ) + f ( n ; i − 1 , j − 1) , for all n, i, j ≥ 0. T ogether with the b oundary co nditions f ( n ; − 1 , 0) = f ( n ; 0 , − 1) = 0 ( n ≥ 0) and f (0; i, j ) = δ i,j, 0 ( i, j ≥ 0), this recurrence eq uation implies the functional equation F ( t ; x, y ) = 1 +  1 x + 1 y + xy  tF ( t ; x, y ) − 1 y tF ( t ; x, 0) − 1 x tF ( t ; 0 , y ) 4 ALIN BOST AN AND MANUEL KAUERS for the generating function F ( t ; x, y ) = ∞ X n =0  ∞ X i,j =0 f ( n ; i, j ) x i y j  t n . Noting that F ( t ; 0 , y ) and F ( t ; y , 0) ar e equal by the s y mmetry of the step set ab out the main diagonal of N 2 , the last eq uation b ecomes F ( t ; x, y ) = 1 +  1 x + 1 y + xy  tF ( t ; x, y ) − 1 y tF ( t ; x, 0) − 1 x tF ( t ; y , 0) . A t the hea r t o f our next arguments is the kernel met ho d , a metho d co mmonly a ttributed to Knuth [ 17 , Solutions of Exercis e s 4 and 11 in § 2.2.1 ] w hich has alrea dy b een used to great a dv a nt age in la ttice path counting, see e.g. [ 12 , 27 , 6 ]. After bringing the functional equation for F ( t ; x, y ) to the form (( x + y + x 2 y 2 ) t − xy ) F ( t ; x, y ) = xtF ( t ; x, 0 ) + y tF ( t ; y , 0) − xy , (K) the kernel metho d co nsists of co upling x and y in s uch a wa y that this equation reduces to a simpler one, from which useful information ab out the se ction series F ( t ; x, 0) can b e extracted. In the present case, the substitution y → Y ( t, x ) = x − t − √ − 4 t 2 x 3 + x 2 − 2 tx + t 2 2 tx 2 = t + 1 x t 2 + x 3 +1 x 2 t 3 + 3 x 3 +1 x 3 t 4 + 2 x 6 +6 x 3 +1 x 4 t 5 + · · · ∈ Q [ x, x − 1 ][[ t ]] , which is leg itimate s ince the p ow er ser ies Y ( t, x ) has p ositive v aluation, puts the left hand s ide of ( K ) to zero, and therefore shows tha t U = F ( t ; x, 0) is a so lution of the r e duc e d kernel e quation (K red ) U ( t, x ) = Y ( t, x ) t − Y ( t, x ) x U ( t, Y ( t, x )) . Now, the key feature of Equation ( K red ) is that its unique solution in Q [[ x, t ]] is U = F ( t ; x, 0 ). This is a consequence of the following eas y le mma. Here, and in the res t of the ar ticle, o r d v S denotes the v a luation of a p ower series S with resp ect to some v aria ble v o ccurr ing in S . Lemma 4. L et A, B , Y ∈ Q [ x, x − 1 ][[ t ]] b e such that o rd t B > 0 and or d t Y > 0 . Then ther e ex ists at most one p ower series U ∈ Q [[ x, t ]] with U ( t, x ) = A ( t, x ) + B ( t, x ) · U ( t, Y ( t, x )) . Pr o of. By linear it y , it suffices to show that the only solution in Q [[ x, t ]] of the homogeneous equation U ( t, x ) = B ( t, x ) · U ( t, Y ( t, x )) is the trivial solution U = 0. This is a direct co ns equence of the fact that if U were non-zer o , then the v aluation of B ( t, x ) · U ( t, Y ( t, x )) w ould b e at lea st equal to ord t B + or d t U , thu s strictly gre ater than the v aluation of U ( t, x ), a contradiction.  W e are now ready to repr ov e the following clas sical result. Theorem 5. [ 14 ] F ( t ; x, y ) is algebr aic. Pr o of. The strategy is to us e a computer-a s sisted pro of, w hich is completed in t wo steps : (1) Guess an alg ebraic equatio n for the serie s F ( t ; x, 0), by insp ectio n o f its initial terms . (2) Pr ove that (a) the equatio n gues sed a t Step ( 1 ) admits exactly one s olution in Q [[ x, t ]], denoted F cand ( t ; x, 0 ); (b) the power serie s U = F cand ( t ; x, 0 ) s a tisfies ( K red ). Once this has b een accomplished, the fact that U = F ( t ; x, 0) also satisfies Equation ( K red ), in conjunction with Lemma 4 (with the choice A ( t, x ) = Y ( t, x ) /t and B ( t, x ) = − Y ( t, x ) /x ), implies that the p ow er s eries F cand ( t ; x, 0 ) a nd F ( t ; x, 0) coincide. In par ticular, F ( t ; x, 0) sa tis fie s the gues s ed equation, and this cer tifies that F ( t ; x, 0 ) is an algebraic p ower s e ries. Since Y ( t, x ) is alg ebraic a s well, and since the cla ss of a lgebraic p ower series is clo sed under addition, multiplication and inv ersion, it follows fr o m ( K ) that F ( t ; x, y ) is algebraic , too . This concludes the pro of.  THE COMPLETE GENERA TING FUNCTION FOR GESSEL W ALKS IS ALGEBRAIC 5 In the rest of this section, we supply full details on the automated guessing step ( 1 ) and on the proving steps ( 2a ) and ( 2b ). 2.1. Gues sing. Giv en the fir s t few ter ms of a p ow er series, it is po ssible to de ter mine p otential equations that the p ower series may satisfy , for exa mple b y ma k ing a suita ble ans atz with undeter- mined co efficients and solving a linear system. In pr actice, either Gaussian elimination, or faster, sp ecial pur po se a lgorithms bas ed on Hermite-Pad ´ e approximation [ 1 ], ar e used. The computation of such candidate e q uations is known as automate d guessing and is one of the most widely known features o f pack ages such as Maple ’s gfun [ 29 ]. If s ufficiently many terms of the ser ies are provided, a uto mated guessing will eventually find an equation whenever ther e is one. The metho d has tw o po ssible drawbac ks. First, it may in principle return false equa tions (although, if applied pr op erly , it vir tually never do es so in practice). This is why – in order to provide fully rigoro us pro ofs – equations discov ered by this metho d must b e subsequently prov en by a n indep endent arg ument. Second, if the precision needed to r ecov er the equations is very high, the guessing computations could take extremely long when using traditio na l softw are. This is typically the ca se in the Gessel example treated in Section 3 , for which dedicated, very efficient, algo rithms are needed. In the Kreweras case, the computatio ns a r e feasible in M aple . W e now provide commented co de which enables the discovery of a n alg ebraic equation p otentially satisfied by F ( t ; x, 0). First, a function f is defined which co mputes the num bers f ( n ; i, j ) via the multiv aria te recurrence ( 1 ). > f:=pro c(n,i ,j) option rememb er; if i<0 or j<0 or n<0 then 0 elif n=0 then if i=0 and j=0 then 1 else 0 fi else f(n-1 ,i-1, j-1)+f (n-1,i,j+1)+f(n-1,i+1,j) fi end: Using this function, we compute the first 80 co efficients of F ( t ; x, 0); they are p olynomials in x with integer co efficients. The r esulting trunca ted p ow er series is stor ed in the v a riable S . > prec:= 80: > S:=ser ies(a dd(ad d(f(k,i,0)*x^i,i=0..k)*t^k,k=0..prec),t,prec-1): Next, starting fr o m S , the gfun g uessing function s eriestoalgeq discov ers a candidate for an al- gebraic e q uation satisfied by F ( t ; x, 0 ). F or efficiency r easons, we do not use the built-in version of gfun , but a recent one which can b e downloaded fr om http:/ /algo .inria.fr/libr aries/papers/gfun.html > gfun:- serie stoal geq(S,Fx(t)): > P:=col lect( numer (subs(Fx(t)=T,%[1])),T); The g uessed p olynomial reads: P ( T , t, x ) = (16 x 3 t 4 + 10 8 t 4 − 72 xt 3 + 8 x 2 t 2 − 2 t + x ) + (96 x 2 t 5 − 48 x 3 t 4 − 14 4 t 4 + 10 4 xt 3 − 16 x 2 t 2 + 2 t − x ) T + (48 x 4 t 6 + 19 2 xt 6 − 26 4 x 2 t 5 + 64 x 3 t 4 + 32 t 4 − 32 xt 3 + 9 x 2 t 2 ) T 2 + (192 x 3 t 7 + 12 8 t 7 − 96 x 4 t 6 − 19 2 xt 6 + 128 x 2 t 5 − 32 x 3 t 4 ) T 3 + (48 x 5 t 8 + 19 2 x 2 t 8 − 19 2 x 3 t 7 + 56 x 4 t 6 ) T 4 + (96 x 4 t 9 − 48 x 5 t 8 ) T 5 + 16 x 6 t 10 T 6 . Running Maple 12 o n a moder n laptop 1 , the whole guessing computation requir es ab out 80Mb of memor y a nd takes less than 20 seconds. O nce the ca ndidate p oly nomial P is gues s ed, one could pro ceed to its empiric al certification; this can be done in v arious ways, as explained in [ 5 ]. W e do not nee d to do this here, since we a r e going to pr ove in § 2.2 that F ( t ; x, 0 ) is a ro ot of P . 1 MacBook Pro; Int el Core 2 Duo Pr o cessor, @2.4 GHz; 4Mb cac he, 2Gb R A M. 6 ALIN BOST AN AND MANUEL KAUERS One may wonder wher e the precision 8 0 used in the previous co mputations comes from. Here, this precisio n was humanly g uessed, b eing chosen as a reasona ble thresho ld. How ev er, a straight- forward do ubling technique (not explained here in detail) would allow to automatic al ly tune it, by running several times the whole guessing pro cedure with increa sing pr e c ision until the same po lynomial is output tw o consecutive times. 2.2. Proving. In this section, we deta il the tw o s teps ( 2a ) and ( 2b ) used in the pro o f of Theor e m 5 . 2.2.1. Existenc e and Uniqueness. Since P (1 , 0 , x ) = 0 and ∂ P ∂ T (1 , 0 , x ) = − x , the implicit function theorem implies that P admits a unique ro ot F cand ( t ; x, 0 ) in Q (( x ))[[ t ]]. It follows that P has at most one ro ot in Q [[ x, t ]] and tha t this ro ot, if it ex is ts, b elo ngs to Q [ x, x − 1 ][[ t ]]. Proving the existenc e of a ro ot of P in Q [[ x, t ]] is less straig ht forward: this time, the equalities P (1 , 0 , 0) = 0 and ∂ P ∂ T (1 , 0 , 0) = 0 preven t us fro m directly inv oking the implicit function theo r em. W e are thus faced to a clumsy technical complica tion, since what we rea lly nee d to prov e is that the ro ot F cand ( t ; x, 0 ) actually b elong s to Q [[ x, t ]]: other wise, the substitution of U = F cand ( t ; x, 0 ) in E q uation ( K red ), used in Step ( 2b ) o f the pro of of Thm. 5 , would not b e legitimate. T o cir cumv en t this complication, we exploit the fact that, when seen in Q ( x )[ T , t ], the p olyno- mial P ( T , t, x ) defines a curve o f g enus zero over Q ( x ), which can thus b e r a tionally para meter ized. Precisely , using Map le ’s algcurves pack age, the rational functions R 1 ( U, x ) and R 2 ( U, x ) defined by: R 1 ( U, x ) = U (1 + U )(1 + 2 U + U 2 + U 2 x ) 2 h ( U, x ) , R 2 ( U, x ) = ( U 4 x 2 + 2 U 2 ( U + 1) 2 x + 1 + 4 U + 6 U 2 + 2 U 3 − U 4 ) h ( U, x ) (1 + U ) 2 (1 + 2 U + U 2 + U 2 x ) 4 , with h ( U, x ) = U 6 x 3 + 3 U 4 ( U + 1) 2 x 2 + 3 U 2 ( U + 1) 4 x + 1 + 6 U + 15 U 2 + 24 U 3 + 27 U 4 + 18 U 5 + 5 U 6 , are found to s ha re the following pr op erties: • P ( R 2 ( U, x ) , R 1 ( U, x ) , x ) = 0; • there exists a (unique) p ower serie s U 0 ( t, x ) = t + t 2 + ( x + 1) t 3 + (2 x + 5) t 4 + (2 x 2 + 3 x + 9) t 5 + . . . in Q [[ x, t ]] such that R 1 ( U 0 , x ) = t and U 0 (0 , x ) = 0. While the first prop erty is ea sily chec k ed by dir ect calcula tion, the seco nd one is a conse- quence of the implicit function theorem, since Q ( U, t, x ) = R 1 ( U, x ) − t satisfies Q (0 , 0 , 0) = 0 a nd ∂ Q ∂ U (0 , 0 , 0) = 1. The existence pr o of of a p ower series s o lution of P is then co mpleted using the following argument: R 2 having no po le at U = 0, and the v alua tio n o f U 0 with resp ect to t be ing p os itive, the co mpo sed p ow er series R 2 ( U 0 ( t, x ) , x ) is well defined in Q [[ x, t ]] and it satisfie s P ( R 2 ( U 0 , x ) , t, x ) = P ( R 2 ( U 0 , x ) , R 1 ( U 0 , x ) , x ) = 0 . Therefore, F cand ( t ; x, 0 ) = R 2 ( U 0 ( t, x ) , x ) is the unique p ow er series solution in Q [[ x, t ]] of P . 2.2.2. Comp a tibility with the r e d uc e d kernel e quation. W e need to show that F cand ( t ; x, 0 ) so de- fined satisfies equatio n ( K red ). This can b e do ne in v a r ious wa ys by res orting to closur e pro p er ties for algebr aic p ow er series. Thes e closure prop er ties are p erformed by means of resulta nt compu- tations, bas e d on Lemma 6 b elow. One p oss ibility is to first prov e that the p ower s e ries S ( t, x ) ∈ Q [ x, x − 1 ][[ t ]] defined by S ( t, x ) = Y ( t, x ) t − Y ( t, x ) x F cand ( t ; Y ( t, x ) , 0) is a ro o t of the p olynomia l P ( T , t, x ), and then to use the fact that P has only one ro ot in Q [ x, x − 1 ][[ t ]], namely F cand ( t ; x, 0 ). This will imply that S ( t, x ) and F cand ( t ; x, 0 ) coincide, and th us tha t F cand ( t ; x, 0 ) s a tisfies equa tion ( K red ), as desired. THE COMPLETE GENERA TING FUNCTION FOR GESSEL W ALKS IS ALGEBRAIC 7 The main p o int of this appr oach is that, s ince the p ow er s e r ies Y ( t, x ) and F cand ( t ; x, 0 ) are bo th alge braic, finding a p olynomial which annihilates the ser ies S ( t, x ) can be done in an exact manner, without having to a ppe al to gues sing routines. Moreov er, the minimal p o lynomial o f S ( t, x ) ca n b e determined by factoring an annihilating p oly nomial obtained thro ugh a resultant computation, a nd, if neces sary , by matching the ir reducible factor s a gainst the initia l terms of the series S ( t, x ). More precisely , one can use the following classic a l fa c ts, that we re call for completeness, see e.g. [ 20 ] for a pro of. Lemma 6 . L et K b e a field and let P , Q ∈ K [ T , t, x ] b e annihilating p olynomials of two algebr aic p ower series A, B in K [ x, x − 1 ][[ t ]] . Then (1) pA is algebr aic for every p ∈ K ( t, x ) , and it is a r o ot of p deg T P P ( T /p , t, x ) . (2) A ± B is algebr aic, and it is a r o ot of res z ( P ( z , t, x ) , Q ( ± ( T − z ) , t, x )) . (3) AB is algebr aic, and it is a r o ot of res z ( P ( z , t, x ) , z deg T Q Q ( T /z , t, x )) . (4) If or d x B > 0 , then A ( t, B ( t, x )) is algebr aic, and it is a r o ot of r e s z ( P ( T , t, z ) , Q ( z , t, x )) . Since z /t − z /xF cand ( t ; z , 0) is a ro ot o f (the nu merator of ) P ( x/z ( z /t − T ) , t, z ) and since Y ( t, x ) is a r o ot of ( x + T + x 2 T 2 ) t − xT , Lemma 6 sugges ts contin uing our M aple sessio n by co nstructing a p olyno mia l in Q [ T , t, x ] which has S ( t, x ) as a ro ot, in the following wa y: > ker := (T,t,x ) -> (x+ T+x^2 *T^2) *t-x*T: > pol := unappl y(P,T ,t,x) : > res := result ant(n umer( pol(x/z*(z/t-T),t,z)), ke r(z,t ,x), z): > factor (prim part( res,T)); The o utput of the last line is P ( T , t, x ) 2 , which prov es that S ( t, x ) is a ro ot of P ( T , t, x ). 2.3. Cons equences. Setting x to 0 in P leads to the conclus ion that the g enerating series F ( t ; 0 , 0) of Kr e weras excur sions is a r o ot of the p oly nomial 64 t 6 T 3 + 16 t 3 T 2 + T − 72 t 3 T + 54 t 3 − 1. An argument simila r to that used in the pro of of Coro llary 2 then implies that the co efficients a n of F ( t ; 0 , 0) satisfy the linear r ecursion ( n + 6 )(2 n + 9) a n +3 − 54 ( n + 2)( n + 1) a n = 0 , a 0 = 1 , a 1 = 0 , a 2 = 0 , which in turn provides an a lternative pro of of the classica l fact [ 19 , 1 4 , 6 ] that the ser ies F ( t ; 0 , 0) is b oth alge br aic a nd hype rgeometric , a nd it has the following clo sed for m F ( t ; 0 , 0) = 3 F 2  1 / 3 2 / 3 1 3 / 2 2     27 t 3  = ∞ X n =0 4 n  3 n n  ( n + 1 )(2 n + 1) t 3 n . 3. Gessel w alks F or esta blis hing the pro of of Theor em 3 , we apply e s sentially the same r easoning that w as applied in the prev io us section for proving Theo r em 5 . The main difference is that the intermediate expressions get very big, so that they ca n only b e handled by sp ecial purp os e softw are (see the data provided on our website [ 4 ]). Ther e are also some additional co mplications which re q uire to v ar y the arguments slightly . In this section, we p oint out these complicatio ns, describ e how to circumv en t them, and w e do cument our computations. The num bers g ( n, i , j ) of Gessel walks o f leng th n ending at ( i , j ) ∈ Z 2 satisfy the recurrence equation g ( n + 1; i, j ) = g ( n ; i − 1 , j − 1) + g ( n ; i + 1 , j + 1) + g ( n ; i − 1 , j ) + g ( n ; i + 1 , j ) for n , i, j ≥ 0. T ogether with a ppr opriate b o undary conditions, this equa tion implies that the generating function G ( t ; x, y ) = ∞ X n =0  ∞ X i,j =0 g ( n ; i, j ) x i y j  t n , which we seek to prove algebra ic, satisfies the equation ((1 + y + x 2 y + x 2 y 2 ) t − xy ) G ( t ; x, y ) = (1 + y ) t G ( t ; 0 , y ) + t G ( t ; x, 0) − t G ( t ; 0 , 0) − xy . (K G ) 8 ALIN BOST AN AND MANUEL KAUERS This is the s ta rting p oint for the kernel metho d. In this cas e, b ecause of lack o f symmetry with r esp ect to x and y , there a re tw o different wa ys to put the left hand side to zero, using the tw o substitutions y → Y ( t, x ) := −  tx 2 − x + t + p ( tx 2 − x + t ) 2 − 4 t 2 x 2  (2 tx 2 ) = 1 x t + x 2 +1 x 2 t 2 + x 4 +3 x 2 +1 x 3 t 3 + x 6 +6 x 4 +6 x 2 +1 x 4 t 4 + · · · and x → X ( t, y ) :=  y − p y ( y − 4 t 2 ( y + 1) 2 )  (2 ty ( y + 1 )) = y +1 y t + ( y +1) 3 y 2 t 3 + 2( y +1) 5 y 3 t 5 + 5( y +1) 7 y 4 t 7 + · · · They yield the e q uations G ( t ; x, 0 ) = xY ( t, x ) /t + G ( t ; 0 , 0 ) − (1 + Y ( t, x )) G ( t ; 0 , Y ( t, x )) , (1 + y ) G ( t ; 0 , y ) = X ( t, y ) y /t + G ( t ; 0 , 0) − G ( t ; X ( t, y ) , 0) , (K G red ) resp ectively . Note that the first eq ua tion is free o f y while the second is free of x . If we rename y to x in the se c ond equation, then all quantities belong to Q [ x, x − 1 ][[ t ]]. Note also that we can write G ( t ; x, 0) = G ( t ; 0 , 0) + xU ( t, x ) and G ( t ; 0 , x ) = G ( t ; 0 , 0) + xV ( t, x ) for ce rtain p ower ser ie s U, V ∈ Q [[ x, t ]]. In terms of U a nd V , the tw o equations a b ove are then equiv alen t to xU ( t, x ) = xY ( t, x ) /t − (1 + Y ( t, x )) G ( t ; 0 , 0) − Y ( t, x )(1 + Y ( t, x )) V ( t, Y ( t, x )) , (1 + x ) xV ( t, x ) = X ( t, x ) x/t − (1 + x ) G ( t ; 0 , 0 ) − X ( t, x ) U ( t, X ( t, x )) . (K G , 2 red ) The t w o equa tions ( K G , 2 red ) co rresp o nd to the equation ( K red ) in Section 2 . The s ituation here is more complicated in tw o r esp ects. First, we have tw o eq uations and tw o unknown p ow er series U and V ra ther tha n a single equatio n with a single unknown power s eries F ( t ; x, 0); this difference originates from the lack of sy mmetr y of G ( t ; x, y ) with re s p ect to x a nd y , which itself comes from the asymmetry of the Ges sel step set with resp ect to the main diagona l of N 2 . Sec o nd, the tw o equations for U and V still contain G ( t ; 0 , 0) while there is no ter m F ( t ; 0 , 0) prese nt in ( K red ); this difference originates fro m the fact that Gessel’s step se t contains the admissible step ւ , as opp osed to Kre weras’s step set. The o ccurrence of G ( t ; 0 , 0 ) in the equations ( K G , 2 red ) is not really problematic, as we know this p ow er ser ie s explicitly thanks to Theor em 1 . As for the other difference, w e need the following v a riation o f Lemma 4 . Lemma 7. L et A 1 , A 2 , B 1 , B 2 , Y 1 , Y 2 ∈ Q [ x, x − 1 ][[ t ]] b e such that or d t B 1 > 0 , ord t B 2 > 0 , ord t Y 1 > 0 and ord t Y 2 > 0 . Then ther e exists at most one p ai r ( U 1 , U 2 ) ∈ Q [[ x, t ]] 2 with U 1 ( t, x ) = A 1 ( t, x ) + B 1 ( t, x ) · U 2 ( t, Y 1 ( t, x )) , U 2 ( t, x ) = A 2 ( t, x ) + B 2 ( t, x ) · U 1 ( t, Y 2 ( t, x )) . Pr o of. By linearity , it suffices to show that the only solutio n ( U 1 , U 2 ) in Q [[ x, t ]] × Q [[ x, t ]] of the homogeneous s y stem U 1 ( t, x ) = B 1 ( t, x ) · U 2 ( t, Y 1 ( t, x )) , U 2 ( t, x ) = B 2 ( t, x ) · U 1 ( t, Y 2 ( t, x )) is the trivial solution ( U 1 , U 2 ) = (0 , 0). This is a dir ect consequence of the fact that if b oth U 1 and U 2 were non-zero, then the v aluation of B 1 ( t, x ) · U 2 ( t, Y 1 ( t, x )) would b e s trictly greater than the v alua tion of U 2 ( t, x ), a nd the v aluation of B 2 ( t, x ) · U 1 ( t, Y 2 ( t, x )) would b e strictly g r eater than the v aluation of U 1 ( t, x ), th us or d t ( U 1 ) > ord t ( U 2 ) > ord t ( U 1 ), a co nt radiction. Ther efore, one of U 1 , U 2 is z e ro, and the sy stem then implies that b oth are zer o.  By a s lightly more careful analysis , the lemma co uld b e refined further such as to show that there is only one tr iple o f p ow er ser ie s ( U, V , G ) with U , V ∈ Q [[ x, t ]] and G ∈ Q [[ t ]] (free of x ) which satisfies ( K G , 2 red ) with G ( t ; 0 , 0) replaced b y G . In this version, the pro of co uld be completed without r eference to the indep endent pr o of of Thm. 1 . Either wa y , we ca n in principle pro ceed from this p o int as in Sectio n 2 . O ut o f conv enience, we choose to rega r d G ( t ; 0 , 0) as known. Again, w e divide the r emaining tas k in tw o steps : THE COMPLETE GENERA TING FUNCTION FOR GESSEL W ALKS IS ALGEBRAIC 9 (1) Guess defining alg ebraic equatio ns fo r U ( t, x ) and V ( t, x ), by insp ecting the initial terms of G ( t ; x, 0), resp. o f G ( t ; 0 , x ). (2) Pr ove that (a) ea ch of the g ues sed equations has a unique solution in Q [[ x, t ]], denoted U cand ( t ; x, 0 ), resp. V cand ( t ; x, 0 ); (b) the power serie s U cand and V cand indeed sa tisfy the t wo equa tions in ( K G , 2 red ). Once this ha s bee n acco mplished, Lemma 7 implies that the candida te s eries ar e actually equa l to U a nd V , r e sp ectively , and so these ser ies a s well a s G ( t ; x, 0 ) and G ( t ; 0 , y ) ar e in particula r algebraic . Then equation ( K G ) implies that G ( t ; x, y ) is alg ebraic, to o. This then completes the pro of o f Thm. 3 . 3.1. Gues sing. In the b eginning, we had no reason to sus pe c t tha t G ( t ; x, y ) is alg ebraic, since even the sp ecializatio n G ( t ; 0 , 0) was genera lly thought to b e transcendental. Motiv a ted by the case x = y = 0 (i.e., by Thm. 1 , which was merely a co njectur e by that time), we want ed to find out whether G ( t ; x, y ) has chances to b e D-finite with resp ect to t , a nd searched for linear differential eq ua tions with p olynomial co efficients p otentially sa tisfied by its sections G ( t ; x, 0) and G ( t ; 0 , y ). With such equa tions at hand, w e co uld hav e, in principle, proven the D-finiteness of G ( t ; x, y ), b y very muc h the same re a soning that we apply here for proving that G ( t ; x, y ) is alg ebraic. W e rea lized quickly that the different ial equations for G ( t ; x, 0) and G ( t ; 0 , y ), if they exist, a re to o big to b e caught by the g uessers implemen ted in pack a g es like Maple ’s gfun or Mathem atica ’s GeneratingFunctions . In order to gain efficiency , we switched to Magma , which provides efficie nt implemen tations o f low-lev el a lgorithms, a nd we opted for applying a mo dular approach: we set x and y to s p e c ial v a lues x 0 , y 0 = 1 , 2 , 3 , . . . , and in addition, we kept numerical co efficients r educed mo dulo several fixed pr imes p to av oid the emerging of lar ge r ational num bers. Mo dulo a pr ime p , and starting fr om the first 1000 ter ms o f the s eries G ( t ; x, 0) and G ( t ; 0 , y ), we used a very efficient automated guessing scheme, r elying o n the Beckermann-Labahn (FFT-based) sup er-fas t algor ithm for computing Hermite-Pad´ e approximant s [ 1 ]. E ven tually , we made the following obs erv ations: • F or any choice of p a nd x 0 , there are several differential o pe r ators in Z p [ t ] h D t i , of order 14 and with co efficients o f degree at most 43, which seem to annihila te G ( t ; x 0 , 0) in Z p [[ t ]]. • F or any choice of p and y 0 , there ar e s everal differential o pe rators in Z p [ t ] h D t i , o f order 15 and with co efficients o f degree at most 34, which seem to annihila te G ( t ; 0 , y 0 ) in Z p [[ t ]]. (Here, and herea fter, D t stands for the usual deriv ation o pe rator d dt , and for any ring R , we denote by R [ t ] h D t i the W eyl alg ebra of differential o p e r ators with p olynomial co efficients in t over R .) The nex t idea was to apply a n interpolation mechanism in orde r to reco nstruct, starting fro m guesses for v arious choices of x 0 and y 0 , and mo dulo v ar ious primes p , tw o candidate op erator s: one in Q [ x, t ] h D t i that would annihilate G ( t ; x, 0) in Q [ x ][[ t ]], a nd the o ther one in Q [ y , t ] h D t i , that would annihila te G ( t ; 0 , y ) in Q [ y ][[ t ]]. The ingredients needed to put such an interpola tion scheme into practice ar e r ational function interp olation and r atio nal nu mb er r e c onstruction . Both are s ta ndard techniques in co mputer alg ebra, for details on fa st a lgorithms we refer to [ 13 ]. T o o ur sur prise, when applied to the or der 14 a nd 15 differ ential op erato rs mentioned ab ove, we found this re c onstruction scheme to require an unrea sonably larg e num ber of ev alua tion p o int s x 0 , y 0 = 1 , 2 , 3 , . . . , sugg esting an unreas onably high degre e o f the op era tors with resp ect to x o r y , resp ectively . W e ab orted the computation when the exp ected degree exceeded 150 0 (!). At this po int, we had the impress io n that the s e ction ser ies G ( t ; x, 0) and G ( t ; 0 , y ) might not b e D-finite. Our next a ttempt was to find ca ndidate op erator s o f smaller total size by trading o r der against degree. W e wen t back to the serie s G ( t ; x 0 , 0) mo dulo p , a nd tried to de ter mine the le ast or d er op erator L ( p ) x 0 , 0 ∈ Z p [ t ] h D t i annihilating it. This w as done by taking several candidate op erato r s in Z p [ t ] h D t i of order 14 as ab ove, and by computing their greates t common right diviso r (g crd) in the ra tional W eyl algebra Z p ( t ) h D t i . (Despite the non-commutativit y o f Z p ( t ) h D t i , this can b e done by a v ariant of the Euclidean alg orithm [ 25 , 9 ], see [ 15 ] for a mo re efficient grcd algorithm.) 10 ALIN BOST AN AND MANUEL KAUERS W e applied the same strategy to find the le ast or der op era to r L ( p ) 0 ,y 0 ∈ Z p [ t ] h D t i a nnihilating the series G ( t ; 0 , y 0 ) mo dulo p . Doing so fo r several ev a luation p oints x 0 , y 0 = 1 , 2 , 3 , . . . and several choices of p , it was finally po ssible, by using the interpolation s cheme describ ed ab ov e, to r econstruct fro m the v arious mo d- ular candidate o pe rators L ( p ) x 0 , 0 and L ( p ) 0 ,y 0 , tw o c a ndidates L x, 0 ∈ Q [ x, t ] h D t i and L 0 ,y ∈ Q [ y , t ] h D t i with r e asonable deg rees in x and y . The o p er ators L x, 0 and L 0 ,y are p osted on our website [ 4 ]. The op erator L x, 0 has order 11, degree 96 with res p ec t to t and degree only 7 8 (!) in x . Its longest integer co efficients has only 6 1 decimal digits. The op erato r L 0 ,y is even nicer . Its o rder is 1 1 , its degree is 68 with resp ect to t and just 28 (!) with r esp ect to y . Its long est integer co efficient has 51 decimal digits . The whole pr o cedure for guessing L x, 0 and L 0 ,y to ok less than 2 CP U hours on a mo dern computer running Mag ma v 2 .13 (12) . T o achiev e this sp eed, we greatly b enefited, on the o ne side, from the fas t Magma ’s buit-in p olynomial, integer and mo dular arithmetic, and o n the other side, from our own efficient implementations of several alg orithms (e.g., for Hermite-Pad´ e approximation, for ra tional function interp olation and for grcds). F or instance , to compute L x, 0 we use d 21 primes p of 28 bits each and 1 5 8 distinct integer v alues of x 0 . Modulo each prime p , 150 CPU seconds were eno ugh to co mpute: 158 bundles of four differential op erator s in Z p [ t ] h D t i with or der 14 and with co efficients of degree at most 4 3 (b y Her mite-Pad ´ e appr oximation), 158 op era tors in Z p [ t ] h D t i of order 1 1 and with co efficients of degree at most 9 6 (by right g r cd) and 12 × 9 7 = 1164 ratio na l functions in Z p ( x ) with numerators and deno minators of deg ree at most 78 (by ratio na l in terp olation). At this p o int , the o pe rator contained 97 × 79 × 12 = 91 956 ter ms of the form c i,j,k t i x j D k t ( i ≤ 96 , j ≤ 78 , k ≤ 11), where ea ch c i,j,k ∈ Q was known mo dulo the 21 primes . The consta nt s c i,j,k were recovered by p erfo r ming 91956 r ational num ber reconstr uctions. The who le computation of L x, 0 to ok 55 minutes. Gues sing L 0 ,y using the same metho d was even a little bit faster. The exceptionally small siz e s of L x, 0 and L 0 ,y (in compa rison to the intermediate expr essions) sp eak very muc h in fav or o f their cor rectness. Also, the fact that the op erator s L x, 0 and L 0 ,y verify the following eq ualities in Q [[ t ]]: L x, 0 ( G ( t ; x, 0)) mo d t 1000 = 0 a nd L 0 ,y ( G ( t ; 0 , y )) mo d t 1000 = 0 , provides more empirica l ev ide nc e that L x, 0 and L 0 ,y are indeed a nnihilating op erator s for G ( t ; x, 0) and G ( t ; 0 , y ), r esp ectively . There are a num b er of additional tests whic h can b e p erfor med to exp er imentally susta in the evidence that a guessed differential ope r ator is co rrect (see our pap e r [ 5 ] for a collection of such tests), a nd our op e r ators L x, 0 and L 0 ,y successfully pass all these tests. One of the tests consists of chec king whether the o p erator s L x, 0 and L 0 ,y po ssess an a rithmetic prop erty which is exp ected from the minimal or der o p er ator annihilating a genera ting function like G ( t ; x, 0) a nd G ( t ; 0 , y ), see [ 5 ]. This prop erty , called glob al nilp otency [ 11 ], can b e stated a s follows: for almo st a ny prime num ber p , the or der 11 op erato rs L x, 0 , resp. L 0 ,y , should rig ht - divide the pure p ower D 11 · p t in Z p ( x, t ) h D t i , re s p. in Z p ( y , t ) h D t i . W e check ed that this pro p erty indeed holds for all pr imes p < 100 . W e actually found out that the o p erators L x, 0 and L 0 ,y hav e a stronger prop erty: they even rig ht-divide D p t ; in other terms, they hav e zero p -cur v a ture fo r all the tested primes p . This was the key observ ation which le d us to susp ect tha t G ( t ; x, y ) is alge br aic, for accor ding to a famous conjecture o f Grothendieck [ 28 ], an op erator has z ero p -cur v a ture if and only if it admits a basis of a lg ebraic solutions. The conjecture is still op en (even for se c ond order op erator s), but it is g enerally b elie ved to be true. In either ca s e, so mething interesting is g o ing on: either G ( t ; x, y ) is alge br aic or we hav e found op erator s which very m uch lo ok like c o unterexamples to Grothendieck’s conjecture. W e next searched for p otential p olyno mial equatio ns satisfied by the p ow er serie s U , V ∈ Q [[ x, t ]] defined by G ( t ; x, 0 ) = G ( t ; 0 , 0) + xU ( t, x ) and G ( t ; 0 , x ) = G ( t ; 0 , 0) + xV ( t, x ). W e did not find any using o nly 1 000 terms of those series , but we found s o me starting fro m 1 2 00 terms. Using again g ue s sing techniques ba sed o n fast mo dular Hermite-Pad ´ e approximation, combined with THE COMPLETE GENERA TING FUNCTION FOR GESSEL W ALKS IS ALGEBRAIC 11 an interpola tion scheme, we discov ered tw o p o lynomials P 1 ( T , t, x ) ∈ Q [ T , t, x ] and P 2 ( T , t, y ) ∈ Q [ T , t , y ] which sa tisfy P 1 ( U ( t, x ) , t, x ) = 0 mo d t 1200 and P 2 ( V ( t, y ) , t, y ) = 0 mo d t 1200 . These p oly no mials are po s ted on our website [ 4 ]. The p olynomial P 1 has degree s 24, 44, and 3 2 with resp ec t to T , t , and x , resp ectively , a nd inv olv es integers with no mor e tha n 21 de c ima l digits. The p olynomia l P 2 has degrees 24 , 46 , and 56 with re sp ect to T , t , and y , r esp ectively , a nd inv olv es integers with no more than 27 dec imal digits. Spelled out ex plic itly in this a r ticle, they would b o th to gether fill ab out thirt y pages; they a re how ever muc h smaller tha n the differential op erator s L x, 0 and L 0 ,y , for which five hundred pages would no t b e enoug h! Just like L x, 0 and L 0 ,y , the p olynomials P 1 and P 2 pass a num ber o f heuristic tests which let them app ear plausible. W e are now going to prove that the guess ed p olynomia ls P 1 and P 2 are indeed v alid. 3.2. Proving. Let P 1 ∈ Q [ T , t, x ] and P 2 ∈ Q [ T , t, y ] b e the tw o p olynomia ls p osted on the website to this a rticle [ 4 ]. W e show (i) that these p olyno mials admit unique power series so lutions U cand ( t, x ) and V cand ( t, x ), resp ectively , a nd (ii) that these p ower series satisfy the r educed kernel equations ( K G , 2 red ). 3.2.1. Existenc e and Uniqueness. As in the case of Kreweras’s walks, the implicit function theor em do es not apply to these p olynomials, but unlike in the Kreweras case, an existence pro of us ing a suitable r a tional par a meterization is no t p ossible either, beca use the po lynomials a t hand define curves of p ositive genus, a nd therefore a ra tional par ameterizatio n do es not exis t. In order to obtain a pro of in this situation, we pr o ceeded alo ng the fo llowing lines: • First w e used Theorem 3 .6 of McDonald [ 22 ] to obtain the existence o f a series s olution X p,q ∈ Q c p,q t p x q with c p,q = 0 for a ll ( p, q ) outside a certain ha lfplane H ⊆ Q 2 . • Next, we computed a system o f biv ar iate recurrenc e equations with p olyno mia l co efficients that the co efficients c p,q m ust necessa r ily sa tisfy . This can b e done in principle by s o ft- ware pa ck ages such as Ch yzak’s mgfun [ 10 ] o r K outschan’s Holono micFu nctions.m [ 18 ]. How ev er, for reaso ns of efficiency we used o ur own implementation of the res p e ctive algo- rithms. • The form o f the recurr e nc e s together with the shap e of the halfplane H imply that the co efficients c p,q of any solution can be nonzero only in a finite union of cones v + N u + N w with vertices v ∈ Q 2 and basis vectors u , w ∈ Q 2 that can b e computed explicitly . If c p,q 6 = 0 for so me index ( p, q ) in such a cone , then als o the co efficient at the cone’s vertex m ust b e nonzero. • Applying McDona ld’s gener alization of Puiseux’s a lgorithm, we determined the first co - efficients o f ser ies solutions to an accura c y that all further co efficients be long to some translate of H which contains no vertices. • As one of these partial solutions contained no terms with fractiona l p ow ers, it was p os sible to conclude that the entire ser ies contains no terms with fra ctional exp onents. Refer ence to u and w implied that this partial solution co uld also not co nt ain any terms with neg ative int egral exp onents, so the only rema ining p os sibility was tha t the solution is in fact a power ser ie s. A full description of the a rgument r equires a somewhat lengthy discussion of a num ber of techn ical deta ils, which we prefer to avoid her e. A s upplemen t to this a rticle is provided on our website [ 4 ] in which we carry out existence pro ofs in full detail that b oth P 1 and P 2 admit some power ser ie s solutions U cand and V cand , r esp ectively . 12 ALIN BOST AN AND MANUEL KAUERS 3.2.2. Comp a tibility with t he re duc e d kernel e qu ation. It remains to show that these so lutions U cand and V cand satisfy the system ( K G , 2 red ). Because of X ( t, Y ( t, x )) = x , the substitution x → Y ( t, x ) transforms the second e q uation of that system to the first. Therefore, it suffices to prov e the second equa tion: (2) (1 + x ) xV cand ( t ; x, 0 ) = X ( t, x ) x/t − (1 + x ) G ( t ; 0 , 0 ) − X ( t, x ) U cand ( t ; X ( t, x ) , 0) . Letting G 1 ( t, x ) = G ( t ; 0 , 0) + xU cand ( t ; x, 0 ) and G 2 ( t, x ) = G ( t ; 0 , 0) + xV cand ( t ; x, 0 ), the la st equation is equiv a lent to (3) (1 + x ) G 2 ( t, x ) − G ( t ; 0 , 0) = xX ( t, x ) /t − G 1 ( t, X ( t, x )) . By Cor ollary 2 and Lemma 6 , the p ower se r ies (1 + x ) G 2 ( t, x ) − G ( t ; 0 , 0) and xX ( t, x ) /t − G 1 ( t, X ( t, x )) are algebra ic a nd we can co mpute their minimal po lynomials—at least in theory . Now the p oly- nomials P 1 and P 2 are so big that the requir ed resultant co mputations cannot b e carried out by Maple o r Mathe matica . There ar e efficient sp ecial purp ose algorithms av a ilable for the pa rticular kind o f resulta nt s at hand [ 3 ] and o ur Magma implementation of these alg orithms is able to p er form the necessar y computations. It turns out that the minimal po ly nomials for b oth p ow er ser ies a re identical. It is provided electro nically on the website to this ar ticle. After deter mining a suitable nu mber of initial terms o f b oth s eries and obs erving that they match, it can b e co ncluded that Equations ( 3 ) and ( 2 ) hold. This completes the pro of of Theo rem 3 . 3.3. Cons equences. The fact that G ( t ; x, y ) is a lgebraic has conseq uences which ar e of combi- natorial interest. W e lis t so me. Corollary 8 . The fol lowing series ar e algebr aic: • G ( t ; 1 , 1) – the gener ating function of Gessel walks with arbitr ary endp oint. • G ( t ; 1 , 0) and G ( t ; 0 , 1 ) – the gener ating fun ctions of Gessel walks ending somewher e on the x -axis or the y - axis, r esp e ctively. Using the built-in eq uation solver o f Mathematica, we found that all these series , as well as the series G ( t ; 0 , 0 ), can be express ed in terms of nes ted r adicals, for example G ( t ; 1 , 1) = 1 6 t    − 3 + √ 3 v u u t U ( t ) + s 16 t (2 t + 3 ) + 2 (1 − 4 t ) 2 U ( t ) − U ( t ) 2 + 3    where U ( t ) = q 1 + 4 3 p t (4 t + 1 ) 2 / (4 t − 1) 4 . The radica l repr esentations of the o ther s eries are muc h mor e inv olv ed than this o ne. They are av ailable electronica lly at the website to this article [ 4 ]. Also their minimal po lynomials can b e found there . Corollary 9 . F or every p oint ( i, j ) , the series G i,j ( t ) := P ∞ n =0 g ( n ; i, j ) t n is algebr aic. Pr o of. W e have G i,j ( t ) = 1 i ! j !  d i dx i d j dy j G ( t ; x, y )     x = y =0 and the prop erty of b eing algebra ic is prese r ved under differentiation a nd ev aluation.  The previous co rollar y implies in pa rticular that the conjecture o f Petk o v ˇ sek and Wilf [ 26 ] that g ( n ; 0 , j ) (for fix ed j ) and g ( n ; 1 , 0) ar e P-finite is r ight, a nd that their conjecture that g ( n ; 2 , 0) is not P -finite is wrong. THE COMPLETE GENERA TING FUNCTION FOR GESSEL W ALKS IS ALGEBRAIC 13 F or the degrees of the minimal po lynomials p i,j ( T , t ) o f G i,j ( t ), w e obser ved empirically that deg T p i,j =  4 if i = 2 j + 1 8 otherwis e and deg t p i,j =        12 j − 5 i + 14 if i ≤ j 5 i + 2 j + 14 if j < i < 2 j + 1 6 j + 9 if i = 2 j + 1 7 i − 2 j + 12 if i > 2 j + 1 but we are no t a ble to pr ov e these degree for mulas in gener al. Note that o ur pro of of Theo rem 3 do es no t provide us with the minimal po lynomial of G ( t ; x, y ). This po lynomial will, in fact, b e muc h lar g er than the minimal p olynomia ls o f the ser ies G i,j ( t ) or the series obtained for m G ( t ; x, y ) by setting x , y or t to sp ecial v alues . F r om the s izes of the minimal p olynomia ls of G ( t ; x, 0) and G ( t ; 0 , y ), which we k now explicitly , it can b e deduced that the minimal p olynomia l p ( T , t, x, y ) of G ( t ; x, y ) will hav e degrees 72, 141 , 26 3, a nd 287 with resp ect to T , t , x , a nd y , resp ectively , and thus co nsist of more than 750 Mio terms. Corollary 1 0. G ( t ; x, y ) is D-finite with r esp e ct to any of the variables x, y and t . As every alge braic p ow er ser ies is D-finite, this is an immediate co nsequence of The o rem 3 , even if we rega rd G ( t ; x, y ) as a multiv ariate p ower ser ies in t , x , and y ra ther than as a p ow er s eries in t o nly with x and y b elo nging to the co e fficie nt do main. D-finiteness in t only amounts to the existence of a linea r differential equation in d/dt with co efficients in Q ( x, y )[ t ]. This can b e prov en indep endently in as simila r wa y as Theor em 3 . It suffices to discover differential o p er ators which p o tentially annihilate U ( t, x ) and V ( t, y ), res p ec - tively , define U cand ( t, x ) and V cand ( t, y ) as the unique p ow er ser ie s annihilated by these op erato rs and matching the first terms o f U ( t, x ) and V ( t, y ), resp ectively , a nd prove that these series satisfy the equa tions in ( K G red ). The option of pr oving D-finiteness (with resp ect to t ) dir e c tly is imp orta nt when other step sets instead of Gessel’s {← , → , ր , ւ} are consider e d, fo r which the g enerating function G ( t ; x, y ) is D-finite in t but no t a lg ebraic. As shown by Mishna [ 23 ], such step sets do exist. Explicit knowledge of differential op era tors annihilating G ( t ; x, 0 ) a nd G ( t ; 0 , y ) als o allows to deduce b o unds o n the size o f the differential o p erator annihila ting G ( t ; x, y ). According to a priori estimations, this op er a tors will hav e order up to 22 and poly nomial co efficients with deg rees up to 1968, 9 36, and 336 with resp ect to t , x , and y , resp ectively , and thus consis t o f a b out 1 . 4 · 1 0 10 terms. Corollary 11 . F or fi xe d i and j , the numb er g ( n ; i , j ) c an b e c omp ute d with O ( n ) arithmetic op er ations. F or fi xe d x and y , the c o efficient h t n i G ( t ; x, y ) c an b e c ompute d with O ( n ) arithmetic op er ations. Pr o of. By Cor. 10 , the co e fficient s equence g ( n ; i, j ) is P- finite with r e sp ect to n . Therefore , it satisfies a uniform recurrence with resp ect to n . This rec ur rence, together with appr opriate initial v alues , allows the computatio n o f g ( n ; i 0 , j 0 ) in linear time. 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Algorithms Project, INRIA P aris-R ocquencour t, 781 53 Le Chesna y, France E-mail addr ess : Alin.Bostan@i nria.fr Research Institute for Symb olic Compu t a tion, J. Kepler University Linz, Austria E-mail addr ess : mkauers@risc. uni-linz. ac.at