Diagonals and Walks. Alin Bostan 1 Louis Dumont 1 Bruno Salvy 2. November 4, Introduction Diagonals Walks. 1 INRIA, SpecFun.

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1 and Alin Bostan 1 Louis Dumont 1 Bruno Salvy 2 1 INRIA, SpecFun 2 INRIA, AriC November 4, /16 Louis Dumont and

2 2/16 Louis Dumont and

3 Motivations In combinatorics: Lattice walks w i,j : number of walks that end at (i, j) generating power series: W (X, Y ) = i,j 0 w i,jx i Y j w n,n: number of walks that end on the diagonal at (n, n) diagonal series: diagw (T ) = n 0 w n,nt n also appear in statistical physics, number theory,... /16 Louis Dumont and

4 Motivations In combinatorics: Lattice walks w i,j : number of walks that end at (i, j) generating power series: W (X, Y ) = i,j 0 w i,jx i Y j w n,n: number of walks that end on the diagonal at (n, n) diagonal series: diagw (T ) = n 0 w n,nt n also appear in statistical physics, number theory,... /16 Louis Dumont and

5 F (X, Y ) = 1 1 X Y = i,j 0 diag(f ) = ( ) 2n T n = n n 0 ( ) i + j X i Y j i 1 1 4T Abel ( 1830): Alg D-Finite Furstenberg (1967): Diag = Alg /16 Louis Dumont and

6 1-dimensional walks Set of steps of the form (1, a), a Z Example : Dyck Paths B(T ) = n 0 b nt n, b n : number of bridges of length n B(T ) can be expressed as a diagonal. 6/16 Louis Dumont and

7 Questions Question 1 How hard is it to compute a polynomial equation satised by the diagonal of a bivariate rational function? E(T ) = n 0 e nt n, e n : number of excursions of length n Question 2 How to compute E(T ) mod T N for a given N? 7/16 Louis Dumont and

8 Answers Theorem 1 (B., D., S.) Generically: the minimal polynomial equation can be computed in time that is quasi-linear with respect to its size. the minimal polynomial equation is exponentially bigger than the initial rational function (16 d vs d 2, where d is the degree) d deg T P, deg Y P (2, 2) (16, 6) (108, 20) (640, 70) Theorem 2 (B., D., S.) E(T ) mod T N can be computed in Õ(N) arithmetic operations, with a fairly inexpensive pre-computation. 8/16 Louis Dumont and

9 How to compute a polynomial equation satised by the diagonal of a bivariate rational function? 9/16 Louis Dumont and

10 Algebraic equation for the diagonal Ingredients of the algorithm: Partial fraction decomposition: the diagonal is a sum of residues of a rational function Rothstein-Trager resultant (1976): algebraic equation that cancels all residues α 1 (T ), α 2 (T ),..., α d (T ) Main diculty: if P = d (Y α i=1 i) is known, eciently compute the polynomial P k = (Y (α i1 + α i α ik )) for a given k d {i1,i2,...,i k } 10/16 Louis Dumont and

11 Newton sums Most important part of our algorithm: a way around the main diculty using Newton sums. P = d i=1 (Y α i ) N (P) = n 0 (α n 1 + α n α n d )Y n n! P N (P) is well-known and eective N (P k ) can be computed from N (P) P k is then recovered from N (P k ) 11/16 Louis Dumont and

12 How to compute E(T ) mod T N for a given N? 12/16 Louis Dumont and

13 Excursions: naive method w n,k : number of walks that stay in the upper half plane, of length n and ending at height k e n = w n,0 (e n : number of excursions of length n) w n,k satises a linear recurrence relation with constant coecients : w n,k = w n 1,k a, (1,a) S where S is the set of available steps Algorithm with O(N 2 ) arithmetic operations and no pre-computation 3/16 Louis Dumont and

14 Excursions: more ecient algorithm Idea (Banderier, Flajolet, 2002) Use the fact that E(T ) is algebraic to nd a better recurrence relation. recurrence for e n (1 index) instead of w n,k (2 indices) O(N) operations But linear complexity at the cost of pre-computations: - the algebraic equation, which is exponentially big in d (Bousquet-Mélou, 2008) - algebraic eqn dierential eqn (also exponentially big) (Bostan, Chyzak, Lecerf, Salvy, Schost, 2007) - initial conditions of the recurrence (exponentially many) Algorithm with O(N) operations and pre-computation of an exponential size in d equation 14/16 Louis Dumont and

15 Excursions: new algorithm Ideas Method: E(T ) = T B(t) 1 exp dt (Banderier, Flajolet 2002). Recover 0 t E(T ) from B(T ) using this formula. (including B(T )) satisfy small (polynomial size) dierential equations (Bostan, Chen, Chyzak, Li, 2010)). Fast computation of a dierential equation for B using Hermite reduction (loc. cit.) Calculate B mod T N with this equation O(N) Apply the formula using Newton iteration Algorithm with Õ(N) operations and pre-computation of a polynomial size in d equation 15/16 Louis Dumont and Õ(N)

16 Conclusion 16/16 Louis Dumont and