Fast Computation of the Nth Term of an Algebraic Series over a Finite Prime Field

Transcription

1 Fast Computation of the Nth Term of an Algebraic Series over a Finite Prime Field Alin Bostan 1 Gilles Christol 2 Philippe Dumas 1 1 Inria (France) 2 Institut de mathématiques de Jussieu (France) Specfun Seminar, June 2016 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 1 / 24

2 Motivation Ubiquity: combinatorics, number theory, algebraic geometry Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 2 / 24

3 Motivation Ubiquity: combinatorics, number theory, algebraic geometry Confluence of several domains: functional equations automatic sequences complexity theory Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 2 / 24

4 Motivation Ubiquity: combinatorics, number theory, algebraic geometry Confluence of several domains: functional equations automatic sequences complexity theory One of the most difficult questions in modular computations is the complexity of computations mod p for a large prime p of coefficients in the expansion of an algebraic function. D. Chudnovsky & G. Chudnovsky, 1990 Computer Algebra in the Service of Mathematical Physics and Number Theory Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 2 / 24

5 Problem and main result Input: prime field K = F p E(x, y) K[x, y], with E(0, 0) = 0, E y (0, 0) 0 N N Output: the Nth coefficient of the unique solution f(x) K[[x]] of E(x, f(x)) = 0, f(0) = 0 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 3 / 24

6 Problem and main result Input: prime field K = F p E(x, y) K[x, y], with E(0, 0) = 0, E y (0, 0) 0 N N Output: the Nth coefficient of the unique solution f(x) K[[x]] of E(x, f(x)) = 0, f(0) = 0 Main result: arithmetic complexity linear in log N and almost linear in p Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 3 / 24

7 State of the art First N coefficients complexity at best Θ(N) Chudnovsky + Chudnovsky, 1986 On expansion of algebraic functions in power and Puiseux series, I Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 4 / 24

8 State of the art compute Nth term after precomputation d = deg y E(x, y) Method char. 0 char. p Binary powering (d = 1) O(log 2 N) + O(1) Baby steps Giant steps Õ( N) + O(1) Divide and Conquer O(log p N) Õ(p3d ) Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 5 / 24

9 State of the art compute Nth term after precomputation d = deg y E(x, y) Method char. 0 char. p Binary powering (d = 1) O(log 2 N) + O(1) Baby steps Giant steps Õ( N) + O(1) Divide and Conquer O(log p N) Õ(p3d ) Miller + Brown, 1966 An algorithm for evaluation of remote terms in a linear recurrence sequence Fiduccia, 1985 An efficient formula for linear recurrences Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 5 / 24

10 State of the art compute Nth term after precomputation d = deg y E(x, y) Method char. 0 char. p Binary powering (d = 1) O(log 2 N) + O(1) Baby steps Giant steps Õ( N) + O(1) Divide and Conquer O(log p N) Õ(p3d ) algebraic equation differential equation linear recurrence Chudnovsky + Chudnovsky, 1988 Approximations and complex multiplication according to Ramanujan Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 5 / 24

11 State of the art compute Nth term after precomputation d = deg y E(x, y) Method char. 0 char. p Binary powering (d = 1) O(log 2 N) + O(1) Baby steps Giant steps Õ( N) + O(1) Divide and Conquer O(log p N) Õ(p3d ) algebraic equation Mahler equation DAC recurrence Christol + Kamae + Mendès France + Rauzy, 1980 Suites algébriques, automates et substitutions Allouche + Shallit, 1992 The ring of k-regular sequences Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 5 / 24

12 From algebraic equation to Mahler equation Algebraic equation y = 2x + 2xy + xy 2 + xy 3 1 y y 2 y y 3 y 9 y x x 1+2x 2 +x 3 x 3 1+2x 2 +x 3 +2x 5 +x 6 +x 8 +x 9 +2x 11 +x 12 x 12 p = 3 1+x+2x 2 +x 3 +x 4 x 4 1+x+2x 2 +x 3 +x 4 +2x 5 +2x 6 +2x 7 +x 8 +x 9 +x 10 +2x 11 +x 12 +x 13 x x+2x2 +x 3 x 3 2+2x+x 2 +2x 3 +2x 4 +x 5 +x 6 +x 7 +2x 8 +2x 9 +2x 10 +x 11 +2x 12 x 12 Mahler equation ( 2x 2 + 2x 3 + x 4) f(x) + ( 1 + x 2 + 2x 3 + 2x 4 + x 5 + 2x 6) f(x) 3 + ( 2 + 2x 3 + 2x 5 + x 6 + 2x 9) f(x) 9 + x 9 f(x) 27 = 0 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 6 / 24

13 From algebraic equation to Mahler equation Algebraic equation y = 2x + 2xy + xy 2 + xy 3 1 y y 2 y y 3 y 9 y x x 1+2x 2 +x 3 x 3 1+2x 2 +x 3 +2x 5 +x 6 +x 8 +x 9 +2x 11 +x 12 x 12 p = 3 1+x+2x 2 +x 3 +x 4 x 4 1+x+2x 2 +x 3 +x 4 +2x 5 +2x 6 +2x 7 +x 8 +x 9 +x 10 +2x 11 +x 12 +x 13 x x+2x2 +x 3 x 3 2+2x+x 2 +2x 3 +2x 4 +x 5 +x 6 +x 7 +2x 8 +2x 9 +2x 10 +x 11 +2x 12 x 12 Mahler equation ( 2x 2 + 2x 3 + x 4) f(x) + ( 1 + x 2 + 2x 3 + 2x 4 + x 5 + 2x 6) f(x 3 ) + ( 2 + 2x 3 + 2x 5 + x 6 + 2x 9) f(x 9 ) + x 9 f(x 27 ) = 0 Frobenius f(x) p = f(x p ) Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 6 / 24

14 From Mahler equation to DAC recurrence Algebraic equation y = 2x + 2xy + xy 2 + xy 3 Mahler equation x s f(x q ) ( 2x 2 + 2x 3 + x 4) f(x) + ( 1 + x 2 + 2x 3 + 2x 4 + x 5 + 2x 6) f(x 3 ) + ( 2 + 2x 3 + 2x 5 + x 6 + 2x 9) f(x 9 ) + x 9 f(x 27 ) = 0 Divide-and-conquer recurrence f n s q 2f n 2 + 2f n 3 + f n 4 + f n + f 3 n 2 + 2f n 3 + 2f n 4 + f n 5 + 2f n f n 9 + 2f n f n f n f n f n 9 27 = 0 f x = 0 if x N 0 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 7 / 24

15 From Mahler equation to DAC recurrence Algebraic equation y = 2x + 2xy + xy 2 + xy 3 Mahler equation x s f(x q ) ( 2x 2 + 2x 3 + x 4) f(x) + ( 1 + x 2 + 2x 3 + 2x 4 + x 5 + 2x 6) f(x 3 ) + ( 2 + 2x 3 + 2x 5 + x 6 + 2x 9) f(x 9 ) + x 9 f(x 27 ) = 0 Divide-and-conquer recurrence f n s q 2f n 2 + 2f n 3 + f n 4 + f n + f 3 n 2 + 2f n 3 + 2f n 4 + f n 5 + 2f n f n 9 + 2f n f n f n f n f n 9 27 = 0 f x = 0 if x N 0 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 7 / 24

16 +5x From x algebraic x 129 +x x equation x x x to DAC x 123 +x recurrence 122 +x x x x x x x x x x x x x x x x x x x x 102 +x x x 99 +3x 98 +4x 97 +5x 71 +4x 69 +2x 68 +2x 67 +4x 66 +5x 65 +3x 64 +x 62 +2x 57 +3x 55 +3x 54 +3x 53 +6x 52 +4x 51 +5x 50 +4x 48 +2x 46 +4x 45 +3x 44 +x 43 +6x 42 +5x 41 +2x 40 +5x 39 +3x 38 +6x 37 +3x 36 +2x 35 +x 34 +4x 33 +3x 32 +6x 31 +5x 30 +3x 29 +4x 28 +x 27 +3x 26 +6x 25 +5x 24 +5x 23 +5x 22 +2x 21 +4x 20 +x 18 +2x 17 +5x 16 +5x 15 +3x 14 +4x 13 +6x 12 +2x 11 +4x 10 +2x 9 +x 8 +2x 7 +5x 6 +5x 4 +3x 3 +4x 2 +1)f(x 7 ) +(6x x x x x x x x x x x x 147 +x x x x x x 139 +x x x 135 +x x x x x x x x x x x x x x x x x x x 113 +x x x x x 102 +x x x 98 +2x 71 +3x 69 +x 67 +2x 66 +5x 65 +5x 64 +3x 63 +4x 62 +5x 57 +4x 55 +5x 53 +3x 52 +4x 51 +x 49 +5x 46 +3x 45 +4x 44 +6x 43 +x 42 +2x 41 +5x 39 +3x 38 +4x 37 +5x 36 +x 35 +4x 34 +3x 32 +6x 31 +x 30 +6x 29 +2x 28 +2x 27 +2x 25 +4x 24 +3x 23 +6x 21 +6x 18 +5x 17 +2x 16 +2x 15 +4x 14 +3x 13 +2x 8 +3x 6 +2x 4 +4x 3 +3x 2 +6)f(x 49 ) +(x x x x x x x x x x x 149 +x 147 )f(x 343 )=0 O(log p N) p 3d Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 8 / 24

17 Nth coefficient using section operators Section operators f(x) = n 0 f n x n K[[x]] S r f(x) = k 0 f pk+r x k, 0 r < p Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 9 / 24

18 Nth coefficient using section operators Section operators f(x) = n 0 f n x n K[[x]] S r f(x) = k 0 f pk+r x k, 0 r < p Lemma Let f = n 0 f n x n be in K[[x]] and let N = (N l N 1 N 0 ) p be the radix p expansion of N. Then f N = (S Nl S N1 S N0 f)(0). Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 9 / 24

19 Nth coefficient using section operators N = p = 7 = = (5, 6, 4, 3, 5, 5) 7 f(x) = f 0 + f 1 x + f 2 x 2 + f 3 x 3 + f 4 x 4 + f 5 x 5 + S 5 f(x) = f 5 + f 12 x + f 19 x 2 + f 26 x 3 + f 33 x 4 + f 40 x 5 + S 5 S 5 f(x) = f 40 + f 89 x + f 138 x 2 + f 187 x 3 + f 236 x 4 + f 285 x 5 + S 3 S 5 S 5 f(x) = f f 530 x + f 873 x 2 + f 1216 x 3 + f 1559 x 4 + f 1902 x 5 + S 4 S 3 S 5 S 5 f(x) = f f 3960 x + f 6361 x 2 + f 8762 x 3 + f x 4 + S 6 S 4 S 3 S 5 S 5 f(x) = f f x + f x 2 + f x 3 + f x 4 + S 5 S 6 S 4 S 3 S 5 S 5 f(x) = f f x + f x 2 + f x 3 + S 5 S 6 S 4 S 3 S 5 S 5 f(0) = f Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 10 / 24

20 Diagonal (forward) F (x, y) = y(1 5xy xy2 3xy 3 ) 1 2x 5xy 4xy 2 xy 3 = p = 7 y + 2xy + 3xy 3 + 5xy 4 + 4x 2 y + 3x 2 y 2 + 6x 2 y 4 + 2x 2 y 5 + 2x 2 y 6 + 5x 2 y 7 + x 3 y + 5x 3 y 2 + 3x 3 y 3 + 2x 3 y 5 + 3x 3 y 6 + 6x 3 y 7 + x 3 y 9 + 5x 3 y x 4 y + x 4 y 2 + x 4 y 4 + 5x 4 y 6 + 2x 4 y 8 + x 4 y 9 + 5x 4 y x 5 y + 5x 5 y 2 + 6x 5 y 3 + x 5 y 4 + 6x 5 y 5 + 5x 5 y 7 + 3x 5 y 8 + 3x 5 y 9 + 6x 5 y x 6 y x 5 y x 6 y 12 + x 7 y x 5 y 14 + x 6 y + 2x 6 y 2 + 4x 6 y 3 + 4x 6 y 5 + 5x 6 y 6 + 2x 6 y 8 + 6x 6 y 9 + 5x 6 y x 7 y + 2x 7 y 2 + x 7 y 3 + x 7 y 4 + 5x 7 y 5 + 6x 7 y 6 + 6x 7 y 7 + 6x 7 y 9 + 3x 7 y x 9 y 13 + x 10 y x 9 y x 10 y x 5 y 15 + x 6 y x 7 y 15 + x 8 y x 9 y x 10 y x 5 y x 6 y x 7 y x 8 y + 6x 8 y 3 + 3x 8 y 4 + x 8 y 10 + x 9 y + 6x 9 y 2 + 3x 10 y 2 + 5x 9 y 4 + 4x 9 y 5 + 4x 10 y 5 + 4x 9 y 6 + 6x 10 y 6 + 3x 9 y 7 + 5x 10 y 7 + 3x 8 y 8 + 6x 9 y 8 + 5x 10 y 8 + x 9 y 9 + x 4 y 11 + x 5 y x 6 y x 7 y x 8 y x 9 y 11 + f(x) = DF (x) = 2x + 3x 2 + 3x 3 + x 4 + 6x 5 + 5x 6 + 6x 7 + 3x 8 + x 9 + 4x x 10 y + 6x 10 y 3 + 6x 10 y 9 + 4x 10 y x 9 y x 10 y x 9 y x 10 y 16 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 11 / 24

21 Diagonal (backward) Theorem (Furstenberg s theorem) Every algebraic series is the diagonal of a bivariate rational function. E(x, f(x)) = 0, f(0) = 0, E y (0, 0) 0 f(x) = D a b with a(x, y) = ye y (xy, y), b(x, y) = E(xy, y)/y Furstenberg, 1967, Algebraic functions over finite fields Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 12 / 24

22 Sections are diagonals univariate sections S r : action on algebraic formal series S r f Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 13 / 24

23 Sections are diagonals univariate sections S r : action on algebraic formal series S r f = S r D a b diagonal Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 13 / 24

24 Sections are diagonals univariate sections S r : action on algebraic formal series S r f = S r D a b diagonal bivariate sections S r : action on bivariate rational functions Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 13 / 24

25 Sections are diagonals univariate sections S r : action on algebraic formal series S r f = S r D a b diagonal bivariate sections S r : action on bivariate rational functions S r D a b = DS a r commutation rule b Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 13 / 24

26 Sections are diagonals univariate sections S r : action on algebraic formal series S r f = S r D a b diagonal bivariate sections S r : action on bivariate rational functions S r D a b = DS a r commutation rule b pseudo-sections T r : action on bivariate polynomials Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 13 / 24

27 Sections are diagonals univariate sections S r : action on algebraic formal series S r f = S r D a b diagonal bivariate sections S r : action on bivariate rational functions S r D a b = DS a r commutation rule b pseudo-sections T r : action on bivariate polynomials a S r f = DS r Frobenius b = D S rab p 1 b Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 13 / 24

28 Sections are diagonals univariate sections S r : action on algebraic formal series S r f = S r D a b diagonal bivariate sections S r : action on bivariate rational functions S r D a b = DS a r commutation rule b pseudo-sections T r : action on bivariate polynomials a S r f = DS r b = D S rab p 1 b = D T ra b Frobenius T r v = S r vb p 1 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 13 / 24

29 Sections are diagonals univariate sections S r : action on algebraic formal series S r f = S r D a b diagonal bivariate sections S r : action on bivariate rational functions S r D a b = DS a r commutation rule b pseudo-sections T r : action on bivariate polynomials a S r f = DS r b = D S rab p 1 b Christol, 1979, Ensembles presque périodiques k-reconnaissables = D T ra b Frobenius T r v = S r vb p 1 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 13 / 24

30 Nth coefficient using pseudo-section operators univariate sections S r : action on formal series f N = (S Nl S N1 S N0 f)(0) bivariate sections S r : action on bivariate rational functions pseudo-sections T r : action on bivariate polynomials f N = (T N l T N1 T N0 a)(0, 0) b(0, 0) Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 14 / 24

31 Nth coefficient using pseudo-section operators univariate sections S r : action on formal series f N = (S Nl S N1 S N0 f)(0) bivariate sections S r : action on bivariate rational functions pseudo-sections T r : action on bivariate polynomials f N = (T N l T N1 T N0 a)(0, 0) b(0, 0) Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 14 / 24

32 Finite dimensional stable subspace T r v = S r vb p 1 B = b p 1 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 15 / 24

33 Finite dimensional stable subspace T r v = S r vb p 1 B = b p 1 d x = max(deg x a, deg x b), d y = max(deg y a, deg y b) T r F p [x, y] dx,d y F p [x, y] F p [x, y] v vb p 1 = vb S r vb = T r v Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 15 / 24

34 Finite dimensional stable subspace T r v = S r vb p 1 B = b p 1 d x = max(deg x a, deg x b), d y = max(deg y a, deg y b) T r F p [x, y] dx,d y F p [x, y] F p [x, y] v vb p 1 = vb S r vb = T r v F p [x, y] dx,d y stable by the T r Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 15 / 24

35 A geometrical evidence T r F p [x, y] dx,d y F p [x, y] pdx,pd y F p [x, y] dx,d y v vb p 1 = vb S r vb = T r v p = 11 E = (1 + x)(x y) + x 2 y 2 + (1 + x)y 3 + y 4 a = y xy 2 + 3y 3 + 4y 4 + 3xy 4 + 2x 2 y 4 b = 1+x xy+y 2 +x 2 y+y 3 +xy 3 +x 2 y 3 d x = 2, d y = 4 small box [0, d x] [0, d y] Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 16 / 24

36 A geometrical evidence T r F p [x, y] dx,d y F p [x, y] pdx,pd y F p [x, y] dx,d y v vb p 1 = vb S r vb = T r v dilation with ratio p 1 large box [0, (p 1)d x] [0, (p 1)d y] B = b p 1 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 16 / 24

37 A geometrical evidence T r F p [x, y] dx,d y F p [x, y] pdx,pd y F p [x, y] dx,d y v vb p 1 = vb S r vb = T r v v = xy 3 in the canonical basis translation by (1, 3) very large box [0, pd x] [0, pd y] xy 3 B Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 16 / 24

38 A geometrical evidence T r F p [x, y] dx,d y F p [x, y] pdx,pd y F p [x, y] dx,d y v vb p 1 = vb S r vb = T r v filter r = 3 xy 3 B Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 16 / 24

39 A geometrical evidence T r F p [x, y] dx,d y F p [x, y] pdx,pd y F p [x, y] dx,d y v vb p 1 = vb S r vb = T r v contraction small box [0, d x] [0, d y] T 3 xy 3 = S 3 xy 3 B Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 16 / 24

40 A geometrical evidence T r F p [x, y] dx,d y F p [x, y] pdx,pd y F p [x, y] dx,d y v vb p 1 = vb S r vb = T r v We can compute f N in O((d x d y ) 2 log N) if we know the matrices A 0, A 1,..., A p 1 of T 0, T 1,..., T p 1 in the canonical basis (x n y m ) of F p [x, y] dx,d y. Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 16 / 24

41 Precomputation of A 0, A 1,..., A p 1 All information is in B = b p 1. matrix A r : x n y m x n y m B S r x n y m B translation selection No computation in K = F p, except raising b to the power p 1 Cost: Õ(p 2 ) (binary powering, Kronecker substitution, FFT) Theorem The Nth coefficient can be computed in time Õ(p 2 ) + O(log p N). Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 17 / 24

42 Precomputation of A 0, A 1,..., A p 1 All information is in B = b p 1. matrix A r : x n y m x n y m B S r x n y m B translation selection No computation in K = F p, except raising b to the power p 1 Cost: Õ(p 2 ) (binary powering, Kronecker substitution, FFT) Theorem The Nth coefficient can be computed in time Õ(p 2 ) + O(log p N). To be compared with Õ(p 3d ) O(log p N) Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 17 / 24

43 Only a small part of B = b p 1 is enough Task: Computation of A 0, A 1,..., A p 1 row index i = (k, l) column index j = (n, m) B = α,β c α,β x α y β x n y m x n y m B S r x n y m B translation selection x n y m α,β c α,β x n+α y m+β α,β (C) c α,β x k y l { n + α = pk + r (C) m + β = pl + r = β = α + p(l k) + n m Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 18 / 24

44 Only a small part of B = b p 1 is enough p = 11 p = 109 Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 19 / 24

45 Improved precomputation B(x/t, t) = α,β c α,β x α t β α = δ π δ (x)t δ Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 20 / 24

46 Improved precomputation B(x/t, t) = α,β c α,β x α t β α = δ π δ (x)t δ Frobenius B(x/t, t) = b(x/t, t) p 1 b(x/t, t)p = b(x/t, t) = 1 b(x/t, t) b(xp /t p, t p ) Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 20 / 24

47 Improved precomputation B(x/t, t) = α,β c α,β x α t β α = δ π δ (x)t δ B(x/t, t) = b(x/t, t) p 1 = b(x/t, t)p b(x/t, t) = 1 b(x/t, t) b(xp /t p, t p ) 1 b(x/t, t) = u c u (x)t u rational series Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 20 / 24

48 Improved precomputation B(x/t, t) = α,β c α,β x α t β α = δ π δ (x)t δ B(x/t, t) = b(x/t, t) p 1 = b(x/t, t)p b(x/t, t) = 1 b(x/t, t) b(xp /t p, t p ) 1 b(x/t, t) = u c u (x)t u b(x p /t p, t p ) = v b v(x p )t pv rational series for free Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 20 / 24

49 Improved precomputation B(x/t, t) = α,β c α,β x α t β α = δ π δ (x)t δ B(x/t, t) = b(x/t, t) p 1 = b(x/t, t)p b(x/t, t) = 1 b(x/t, t) b(xp /t p, t p ) 1 b(x/t, t) = u c u (x)t u b(x p /t p, t p ) = v b v(x p )t pv rational series for free π δ (x) = c u (x)b v (x p ) u+pv=δ Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 20 / 24

50 Improved precomputation δ = β α = intercept of the strips with the vertical axis big leaps from one strip to the next Õ(p): Kronecker substitution FFT Newton iteration Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 21 / 24

51 Main result Theorem Let E be in F p [x, y] h,d satisfy E(0, 0) = 0 and E y (0, 0) 0, and let f F p [[x]] be its unique root with f(0) = 0. One can compute the coefficient f N of f in operations in F p. Õ(h(d + h) 5 p) + O(h 2 (d + h) 2 log p N) Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 22 / 24

52 Timings Maple implementation; timings on Intel Core i5, 2.8 GHz, 3MB. running time for N in s. (log scale) p = 9001 d = N (log-log scale) log running time(n) = log log N + C Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 23 / 24

53 Timings Maple implementation; timings on Intel Core i5, 2.8 GHz, 3MB. running time for N in s. (log scale) p = 9001 d = N (log-log scale) log running time(n) = log log N + C running time(n) = K log N Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 23 / 24

54 Timings Maple implementation; timings on Intel Core i5, 2.8 GHz, 3MB. running time for N in s. (log scale) p = 9001 d = N (log-log scale) precomputation time for p in s.(log scale) p (log scale) running time(n) = K log N precomputation time(p) = Kp Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 23 / 24

55 Timings Maple implementation; timings on Intel Core i5, 2.8 GHz, 3MB. running time for N in s. (log scale) p = 9001 d = N (log-log scale) precomputation time for p in s.(log scale) d = p (log scale) running time(n) = K log N precomputation time(p) = Kp log p Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 23 / 24

56 Timings Maple implementation; timings on Intel Core i5, 2.8 GHz, 3MB. running time for N in s. (log scale) p = 9001 d = N (log-log scale) precomputation time for p in s.(log scale) d = p (log scale) O(log N) Õ(p) Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 23 / 24

57 Theorem Let E be in F p [x, y] h,d satisfy E(0, 0) = 0 and E y (0, 0) 0, and let f F p [[x]] be its unique root with f(0) = 0. One can compute the coefficient f N of f in operations in F p. Õ(h(d + h) 5 p) + O(h 2 (d + h) 2 log p N) Thanks for your attention! Bostan, Christol, Dumas Fast Computation of Algebraic Series 41st ISSAC 24 / 24