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. . . . . . . . . . . . Introduction . Mathematical theory Algorithm Efgicient linear computation of the characteristic polynomials of the p -curvatures of a difgerential operator with integer coefgicients. Seminar LFANT Raphal

  1. A then . Mathematical theory . . . . . . . . . Introduction Motivating the p -curvature Algorithm . k is a finite field of characteristic p . Lemma When the connexion on M is of the form A I n A k A k AA k A p Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures M = k ( z ) n can be equipped with the connexion ∂ A : Y �→ Y ′ − AY . ∂ ( f ( z ) · m ) = f ( z ) ∂ · m + f ′ ( z ) · m For all difgerential k ( x ) -module M, m �→ ∂ · m is k ( x p ) -linear. m �→ ∂ p · m is k ( x ) -linear.

  2. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Motivating the p -curvature k is a finite field of characteristic p . Lemma A p Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures M = k ( z ) n can be equipped with the connexion ∂ A : Y �→ Y ′ − AY . ∂ ( f ( z ) · m ) = f ( z ) ∂ · m + f ′ ( z ) · m For all difgerential k ( x ) -module M, m �→ ∂ · m is k ( x p ) -linear. m �→ ∂ p · m is k ( x ) -linear. When the connexion on M is of the form ∂ A then A k +1 = A ′ A 0 = I n k − AA k

  3. . z z z z z z z z z z z z z z and A p z z z z z A z Raphaël Pagès z z z x z x A p z z z z z z z z z z z Motivating the p -curvature . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . Algorithm Mathematical theory Introduction . . . . . . . . . . . . . . . . For ( z + 1) 2 y (3) − zy ′ + ( z 3 + 3) y = 0 and p = 3 .

  4. . z z z z z z z z z z z z z z z and A p z . z z Algorithm A p Raphaël Pagès z z z x z x z z z z z z z z z z Motivating the p -curvature Efgicient computation of p -curvatures Mathematical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . For ( z + 1) 2 y (3) − zy ′ + ( z 3 + 3) y = 0 and p = 3 .   − z 3 +3 0 0 ( z +1) 2 A = 1 0 −   ( z +1) 2   0 1 0

  5. . Algorithm . . . . . . . . . Introduction Mathematical theory Motivating the p -curvature . . z z z A p x z x z z z Raphaël Pagès . Efgicient computation of p -curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . For ( z + 1) 2 y (3) − zy ′ + ( z 3 + 3) y = 0 and p = 3 . 2 z 3     − z 3 +3 0 0 0 ( z +1) 2 ( z +1) 2 ( z +1) 2 2 z 3 2 z 4 +2 z 3 +2 z +1 A =  and A p = 1 0 −     ( z +1) 2 z 3 +1 z 3 +1 ( z +1) 2    2 z 4 z 4 + z 3 + z 2 +2 z +2 2 z 4 +2 z 3 + z +2 0 1 0 z 4 + z 3 + z +1 z 4 + z 3 + z +1 z 3 +1

  6. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Motivating the p -curvature . z z z Raphaël Pagès . Efgicient computation of p -curvatures . . . . . . . . . . . . . . . . . . . . . . . . For ( z + 1) 2 y (3) − zy ′ + ( z 3 + 3) y = 0 and p = 3 . 2 z 3     − z 3 +3 0 0 0 ( z +1) 2 ( z +1) 2 ( z +1) 2 2 z 3 2 z 4 +2 z 3 +2 z +1 A =  and A p = 1 0 −     ( z +1) 2 z 3 +1 z 3 +1 ( z +1) 2    2 z 4 z 4 + z 3 + z 2 +2 z +2 2 z 4 +2 z 3 + z +2 0 1 0 z 4 + z 3 + z +1 z 4 + z 3 + z +1 z 3 +1 z 3 + 1 x + z 6 + 2 z 3 2 χ ( A p ) = x 3 + z 3 + 1

  7. . Algorithm . . . . . . . . . Introduction Mathematical theory Difgerential operators algebra . Definition as sets Example x x x x x x x Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Let A = k [ x ] or k ( x ) with k a field. We define A� ∂ � .

  8. . Mathematical theory . . . . . . . . . . Introduction Algorithm . Difgerential operators algebra Definition Example x x x x x x x Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Let A = k [ x ] or k ( x ) with k a field. We define A� ∂ � . A� ∂ � ≃ A [ ∂ ] as sets

  9. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Difgerential operators algebra Definition Example x x Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Let A = k [ x ] or k ( x ) with k a field. We define A� ∂ � . A� ∂ � ≃ A [ ∂ ] as sets A = Q [ x ] ( x 2 + 2 x + 1) ∂ 3 − x ∂ + ( x 3 + 3)

  10. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Difgerential operators algebra Definition Example Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Let A = k [ x ] or k ( x ) with k a field. We define A� ∂ � . A� ∂ � ≃ A [ ∂ ] as sets A = Q [ x ] ( x 2 + 2 x + 1) ∂ 3 − x ∂ + ( x 3 + 3) ∂ x = x ∂ + 1

  11. A p L : A p L . arithmetic operations. . . . . Introduction Mathematical theory Algorithm summary L is of size O . size : O p . naive computation : O p Best known algorithm : O p arithmetic operations [Bostan, CaRuso, . Schost, 2015]. : size : O . Best known algorithm : O p binary operations [Bostan, CaRuso, Schost, 2014]. Contribution : L x . Computation of all the characteristic polynomials of its p -curvatures for p N in O N binary operations. Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures M = k ( x ) ⟨ ∂ ⟩ / k ( x ) ⟨ ∂ ⟩ L

  12. A p L : A p L . arithmetic operations. . . . . . Introduction Mathematical theory Algorithm summary size : O p . naive computation : O p Schost, 2015]. Best known algorithm : O p arithmetic operations [Bostan, CaRuso, . : size : O . Best known algorithm : O p binary operations [Bostan, CaRuso, Schost, 2014]. Contribution : L x . Computation of all the characteristic polynomials of its p -curvatures for p N in O N binary operations. Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures M = k ( x ) ⟨ ∂ ⟩ / k ( x ) ⟨ ∂ ⟩ L L is of size O (1) .

  13. . arithmetic operations. . . . . . . . Introduction Mathematical theory Algorithm summary naive computation : O p Best known algorithm : O p arithmetic operations [Bostan, CaRuso, . Schost, 2015]. size : O . Best known algorithm : O p binary operations [Bostan, CaRuso, Schost, 2014]. Contribution : L x . Computation of all the characteristic polynomials of its p -curvatures for p N in O N binary operations. Raphaël Pagès . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . M = k ( x ) ⟨ ∂ ⟩ / k ( x ) ⟨ ∂ ⟩ L L is of size O (1) . A p ( L ) : size : O ( p ) . χ ( A p ( L )) :

  14. . summary . . . . . . . . Introduction Mathematical theory Algorithm Best known algorithm : O p arithmetic operations [Bostan, CaRuso, . Schost, 2015]. size : O . Best known algorithm : O p binary operations [Bostan, CaRuso, Schost, 2014]. Contribution : L x . Computation of all the characteristic polynomials of its p -curvatures for p N in O N binary operations. Raphaël Pagès . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . M = k ( x ) ⟨ ∂ ⟩ / k ( x ) ⟨ ∂ ⟩ L L is of size O (1) . A p ( L ) : size : O ( p ) . naive computation : ˜ O ( p 2 ) arithmetic operations. χ ( A p ( L )) :

  15. . Algorithm . . . . . . . . . Introduction Mathematical theory summary . Schost, 2015]. size : O . Best known algorithm : O p binary operations [Bostan, CaRuso, Schost, 2014]. Contribution : L x . Computation of all the characteristic polynomials of its p -curvatures for p N in O N binary operations. Raphaël Pagès . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . M = k ( x ) ⟨ ∂ ⟩ / k ( x ) ⟨ ∂ ⟩ L L is of size O (1) . A p ( L ) : size : O ( p ) . naive computation : ˜ O ( p 2 ) arithmetic operations. Best known algorithm : ˜ O ( p ) arithmetic operations [Bostan, CaRuso, χ ( A p ( L )) :

  16. . Mathematical theory . . . . . . . . . . Introduction Algorithm . summary Schost, 2015]. Best known algorithm : O p binary operations [Bostan, CaRuso, Schost, 2014]. Contribution : L x . Computation of all the characteristic polynomials of its p -curvatures for p N in O N binary operations. Raphaël Pagès . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . M = k ( x ) ⟨ ∂ ⟩ / k ( x ) ⟨ ∂ ⟩ L L is of size O (1) . A p ( L ) : size : O ( p ) . naive computation : ˜ O ( p 2 ) arithmetic operations. Best known algorithm : ˜ O ( p ) arithmetic operations [Bostan, CaRuso, χ ( A p ( L )) : size : O (1) .

  17. . . . . . . . . . . . . Introduction . Mathematical theory Algorithm summary Schost, 2015]. Schost, 2014]. Contribution : L x . Computation of all the characteristic polynomials of its p -curvatures for p N in O N binary operations. Raphaël Pagès . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . M = k ( x ) ⟨ ∂ ⟩ / k ( x ) ⟨ ∂ ⟩ L L is of size O (1) . A p ( L ) : size : O ( p ) . naive computation : ˜ O ( p 2 ) arithmetic operations. Best known algorithm : ˜ O ( p ) arithmetic operations [Bostan, CaRuso, χ ( A p ( L )) : size : O (1) . O ( √ p ) binary operations [Bostan, CaRuso, Best known algorithm : ˜

  18. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm summary Schost, 2015]. Schost, 2014]. Raphaël Pagès . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . M = k ( x ) ⟨ ∂ ⟩ / k ( x ) ⟨ ∂ ⟩ L L is of size O (1) . A p ( L ) : size : O ( p ) . naive computation : ˜ O ( p 2 ) arithmetic operations. Best known algorithm : ˜ O ( p ) arithmetic operations [Bostan, CaRuso, χ ( A p ( L )) : size : O (1) . O ( √ p ) binary operations [Bostan, CaRuso, Best known algorithm : ˜ Contribution : L ∈ Z ( x ) � ∂ � . Computation of all the characteristic polynomials of its p -curvatures for p ⩽ N in ˜ O ( N ) binary operations.

  19. mod p s for all p . Algorithm . . . . . . . . . Introduction Mathematical theory Summary . naive computation : O p binary operations. Best known algorithm : O p binary operations [ChudnovsKy, ChudnovsKy, 1988]. size : O s log p . Best known algorithm : O s p binary operations [StRassen, 1977] . Computation of p N : O sN binary operations [Costa, GeRbicz, HaRvey, 2014]. Raphaël Pagès . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . ( p − 1)! : size : ˜ O ( p ) . ( p − 1)! mod p s :

  20. mod p s for all p . Introduction . . . . . . . . . . Algorithm Mathematical theory . Summary Best known algorithm : O p binary operations [ChudnovsKy, ChudnovsKy, 1988]. size : O s log p . Best known algorithm : O s p binary operations [StRassen, 1977] . Computation of p N : O sN binary operations [Costa, GeRbicz, HaRvey, 2014]. Raphaël Pagès . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . ( p − 1)! : size : ˜ O ( p ) . naive computation : ˜ O ( p 2 ) binary operations. ( p − 1)! mod p s :

  21. mod p s for all p . Introduction . . . . . . . . . . Mathematical theory . Algorithm Summary ChudnovsKy, 1988]. size : O s log p . Best known algorithm : O s p binary operations [StRassen, 1977] . Computation of p N : O sN binary operations [Costa, GeRbicz, HaRvey, 2014]. Raphaël Pagès . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . ( p − 1)! : size : ˜ O ( p ) . naive computation : ˜ O ( p 2 ) binary operations. Best known algorithm : ˜ O ( p ) binary operations [ChudnovsKy, ( p − 1)! mod p s :

  22. mod p s for all p . . . . . . . . . . . . Introduction . Mathematical theory Algorithm Summary ChudnovsKy, 1988]. Best known algorithm : O s p binary operations [StRassen, 1977] . Computation of p N : O sN binary operations [Costa, GeRbicz, HaRvey, 2014]. Raphaël Pagès . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . ( p − 1)! : size : ˜ O ( p ) . naive computation : ˜ O ( p 2 ) binary operations. Best known algorithm : ˜ O ( p ) binary operations [ChudnovsKy, ( p − 1)! mod p s : size : O ( s log ( p )) .

  23. mod p s for all p . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Summary ChudnovsKy, 1988]. Computation of p N : O sN binary operations [Costa, GeRbicz, HaRvey, 2014]. Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . ( p − 1)! : size : ˜ O ( p ) . naive computation : ˜ O ( p 2 ) binary operations. Best known algorithm : ˜ O ( p ) binary operations [ChudnovsKy, ( p − 1)! mod p s : size : O ( s log ( p )) . O ( s √ p ) binary operations [StRassen, 1977] . Best known algorithm : ˜

  24. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Summary ChudnovsKy, 1988]. operations [Costa, GeRbicz, HaRvey, 2014]. Raphaël Pagès . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . ( p − 1)! : size : ˜ O ( p ) . naive computation : ˜ O ( p 2 ) binary operations. Best known algorithm : ˜ O ( p ) binary operations [ChudnovsKy, ( p − 1)! mod p s : size : O ( s log ( p )) . O ( s √ p ) binary operations [StRassen, 1977] . Best known algorithm : ˜ Computation of ( p − 1)! mod p s for all p ⩽ N : ˜ O ( sN ) binary

  25. i f i x x f x Solution : Problem : How to rewrite x ? . Idea : rewrite as elements of k x x x i x x Algorithm Mathematical theory Introduction . p . . k x Raphaël Pagès k k k k k x k x i k x k k x i i f i f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂

  26. i f i x . i . . . Introduction Mathematical theory Algorithm Idea : rewrite as elements of k . Problem : How to rewrite x ? Solution : f x p i i f . f i i k x k k x k x k x k x k k k k Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x

  27. i f i x . i . . . . Introduction Mathematical theory Algorithm Problem : How to rewrite x ? Solution : f x p i i f . f i i k x k k x k x k x k x k k k k Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x Idea : rewrite as elements of k [ θ ] � ∂ � .

  28. i f i x . i . . . . Introduction Mathematical theory Algorithm Problem : How to rewrite x ? Solution : f x p i i f . f i i k x k k x k x k x k x k k k k Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x Idea : rewrite as elements of k [ θ ] � ∂ � .

  29. i f i x . i . . . . . Introduction Mathematical theory Algorithm Problem : How to rewrite x ? f x p i i f . f i i k x k k x k x k x k x k k k k Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x Idea : rewrite as elements of k [ θ ] � ∂ � . Solution : ∂ − 1

  30. . f . . . . . . . Introduction Mathematical theory Algorithm Problem : How to rewrite x ? i f i . i k x k k x k x k x k x k k k k Raphaël Pagès . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x Idea : rewrite as elements of k [ θ ] � ∂ � . Solution : ∂ − 1 f ( x ) = ∑ p − 1 i =0 ( − 1) i f ( i ) ( x ) ∂ − i − 1

  31. . Algorithm . . . . . . . . . Introduction Mathematical theory Problem : How to rewrite x ? . k x k k x k x k x k x k k k k Raphaël Pagès . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x Idea : rewrite as elements of k [ θ ] � ∂ � . Solution : ∂ − 1 f ( x ) = ∑ p − 1 i =0 ( − 1) i f ( i ) ( x ) ∂ − i − 1 ∂ i f ( θ ) = f ( θ + i ) ∂ i

  32. . Introduction . . . . . . . . . . Mathematical theory . Algorithm Problem : How to rewrite x ? k x k x k x k x k k k k Raphaël Pagès . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x Idea : rewrite as elements of k [ θ ] � ∂ � . Solution : ∂ − 1 f ( x ) = ∑ p − 1 i =0 ( − 1) i f ( i ) ( x ) ∂ − i − 1 ∂ i f ( θ ) = f ( θ + i ) ∂ i k ( x ) � ∂ ± 1 � k ( θ ) � ∂ ± 1 �

  33. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Problem : How to rewrite x ? k k k k Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x Idea : rewrite as elements of k [ θ ] � ∂ � . Solution : ∂ − 1 f ( x ) = ∑ p − 1 i =0 ( − 1) i f ( i ) ( x ) ∂ − i − 1 ∂ i f ( θ ) = f ( θ + i ) ∂ i k ( x ) � ∂ ± 1 � k ( θ ) � ∂ ± 1 � k [ x ] � ∂ ± 1 � k [ x ] � ∂ � k ( x ) � ∂ ± 1 � k ( x ) � ∂ �

  34. . . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Problem : How to rewrite x ? Raphaël Pagès . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � θ = x ∂ ∂ ( x ∂ ) = ( x ∂ + 1) ∂ x ( x ∂ ) = ( x ∂ − 1) x Idea : rewrite as elements of k [ θ ] � ∂ � . Solution : ∂ − 1 f ( x ) = ∑ p − 1 i =0 ( − 1) i f ( i ) ( x ) ∂ − i − 1 ∂ i f ( θ ) = f ( θ + i ) ∂ i k ( x ) � ∂ ± 1 � k ( θ ) � ∂ ± 1 � k [ x ] � ∂ ± 1 � k [ θ ] � ∂ ± 1 � k [ x ] � ∂ � k [ θ ] � ∂ � k ( x ) � ∂ ± 1 � k ( θ ) � ∂ ± 1 � k ( x ) � ∂ � k ( θ ) � ∂ �

  35. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm x Example x invertible in k x non invertible in k Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � ∼ k [ x ] � ∂ ± 1 � k [ θ ] � ∂ ± 1 � − → θ∂ − 1 �→ ← � x ∂ θ ∂ ↔ ∂

  36. . . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm x Example Raphaël Pagès . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . k ( θ ) � ∂ ± 1 � et k ( x ) � ∂ ± 1 � ∼ k [ x ] � ∂ ± 1 � k [ θ ] � ∂ ± 1 � − → θ∂ − 1 �→ ← � x ∂ θ ∂ ↔ ∂ ( x + 1) ∂ invertible in k ( x ) � ∂ ± 1 � ∂ + θ non invertible in k ( θ ) � ∂ ± 1 �

  37. B p L l m x l m x p A p L B p L B L B L p Mat l m l m . p . . ... l m l l m l B L . Let L L Raphaël Pagès p i l m i p p . L x x . l x l m B L Algorithm Mathematical theory . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Ξ x ,∂ and Ξ θ,∂ Let L ′ = l m ′ ( θ ) ∂ m + . . . + l 1 ′ ( θ ) ∂ + l 0 ′ ( θ ) .

  38. B p L l m x l m x p A p L B p L p B L B L B L p Mat . . . . ... . Algorithm Mathematical theory Let L l m . x l x . x L p L p i l m i p Raphaël Pagès Introduction Efgicient computation of p -curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ξ x ,∂ and Ξ θ,∂ Let L ′ = l m ′ ( θ ) ∂ m + . . . + l 1 ′ ( θ ) ∂ + l 0 ′ ( θ ) . ′   − l 0 l m ′ ′ 1 − l 1   l m ′ B ( L ′ ) =         ′ − l m − 1 1 l m ′

  39. l m x l m x p A p L B p L ... Let L . . . . m Algorithm Mathematical theory Introduction . . . . x l . l x . x L p L p i l m i p Raphaël Pagès . Efgicient computation of p -curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ξ x ,∂ and Ξ θ,∂ Let L ′ = l m ′ ( θ ) ∂ m + . . . + l 1 ′ ( θ ) ∂ + l 0 ′ ( θ ) . ′   − l 0 l m ′ ′ 1 − l 1   l m ′ B ( L ′ ) =         ′ − l m − 1 1 l m ′ B p ( L ′ ) = Mat ( ∂ p · ) = B ( L ′ )( θ ) B ( L ′ )( θ + 1) . . . B ( L ′ )( θ + p − 1)

  40. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm . ... . . . l m Raphaël Pagès . Efgicient computation of p -curvatures . . . . . . . . . . . . . . . . . . . . . . . . Ξ x ,∂ and Ξ θ,∂ Let L ′ = l m ′ ( θ ) ∂ m + . . . + l 1 ′ ( θ ) ∂ + l 0 ′ ( θ ) . ′   − l 0 l m ′ ′ 1 − l 1   l m ′ B ( L ′ ) =         ′ − l m − 1 1 l m ′ B p ( L ′ ) = Mat ( ∂ p · ) = B ( L ′ )( θ ) B ( L ′ )( θ + 1) . . . B ( L ′ )( θ + p − 1) Let L = l m ( x ) ∂ m + . . . + l 1 ( x ) ∂ + l 0 ( x ) . Ξ x ,∂ ( L ) = l m ( x ) p χ ( A p ( L ))( ∂ p ) ( p − 1 ) Ξ θ,∂ ( L ′ ) = ∏ ′ ( θ + i ) χ ( B p ( L ′ ))( ∂ p ) i =0

  41. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Lemma Send an irreducible element over a power of an irreductible element of the center Multiplicative. Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . Ξ x ,∂ and Ξ θ,∂ Ξ x ,∂ Ξ θ,∂ k [ θ p − θ ][ ∂ p ] k [ x ] � ∂ � k [ x p ][ ∂ p ] k [ θ ] � ∂ � Ξ x ,∂ Ξ θ,∂ k ( θ p − θ )[ ∂ p ] k ( x ) � ∂ � k ( x p )[ ∂ p ] k ( θ ) � ∂ �

  42. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Lemma Send an irreducible element over a power of an irreductible element of the center Multiplicative. Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . Ξ x ,∂ and Ξ θ,∂ Ξ x ,∂ Ξ θ,∂ k [ θ p − θ ][ ∂ p ] k [ x ] � ∂ � k [ x p ][ ∂ p ] k [ θ ] � ∂ � Ξ x ,∂ Ξ θ,∂ k ( θ p − θ )[ ∂ p ] k ( x ) � ∂ � k ( x p )[ ∂ p ] k ( θ ) � ∂ �

  43. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Lemma Send an irreducible element over a power of an irreductible element of the center Multiplicative. Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . Ξ x ,∂ and Ξ θ,∂ Ξ x ,∂ Ξ θ,∂ k [ θ p − θ ][ ∂ p ] k [ x ] � ∂ � k [ x p ][ ∂ p ] k [ θ ] � ∂ � Ξ x ,∂ Ξ θ,∂ k ( θ p − θ )[ ∂ p ] k ( x ) � ∂ � k ( x p )[ ∂ p ] k ( θ ) � ∂ �

  44. . . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Theorem The followin diagram commutes. Raphaël Pagès . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . Ξ x ,∂ and Ξ θ,∂ ∼ k [ x ] � ∂ ± 1 � k [ θ ] � ∂ ± 1 � Ξ x ,∂ Ξ θ,∂ k [ θ p − θ ][ ∂ ± p ] ∼ k [ x p ][ ∂ ± p ]

  45. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm x x x x x x Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Ξ x ,∂ and Ξ θ,∂ ( x 2 + 2 x + 1) ∂ 3 − x ∂ + x 3 + 3

  46. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm x x x x x x Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . Ξ x ,∂ and Ξ θ,∂ ( x 2 + 2 x + 1) ∂ 3 − x ∂ + x 3 + 3 ( ∂ 6 +2 θ∂ 5 +( θ 2 − θ ) ∂ 4 ) ∂ − 3 �→ − ( θ +3) ∂ 3 +( θ 3 − 3 θ 2 +2 θ )

  47. . . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm x Raphaël Pagès . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . Ξ x ,∂ and Ξ θ,∂ ( x 2 + 2 x + 1) ∂ 3 − x ∂ + x 3 + 3 ( ∂ 6 +2 θ∂ 5 +( θ 2 − θ ) ∂ 4 ) ∂ − 3 �→ − ( θ +3) ∂ 3 +( θ 3 − 3 θ 2 +2 θ ) x 3 +3  −  x 2 +2 x +1 1   x 2 +2 x +1 1 0

  48. . . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm . x Raphaël Pagès . Efgicient computation of p -curvatures . . . . . . . . . . . . . . . . . . . . . . Ξ x ,∂ and Ξ θ,∂ ( x 2 + 2 x + 1) ∂ 3 − x ∂ + x 3 + 3 ( ∂ 6 +2 θ∂ 5 +( θ 2 − θ ) ∂ 4 ) ∂ − 3 �→ − ( θ +3) ∂ 3 +( θ 3 − 3 θ 2 +2 θ ) − ( θ 3 − 3 θ 2 + 2 θ )   1 0 x 3 +3    −    x 2 +2 x +1 1 0   1     1 ( θ + 3) x 2 +2 x +1   1 0 − ( θ 2 − θ )   1   1 − 2 θ

  49. x ) with the determinant. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm proof of the commutativity Step 1 : Isomorphism with a matrix algebra afuer scalar extension. Step 2 : The determinant : restriction, corestriction Step 3 : Equility of (resp. Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . M θ M p ( k [ θ p − θ ][ ∂ ± p ][ T ]) ∼ k [ θ ] � ∂ ± 1 � [ T ] ∼ ∼ M x ∼ k [ x ] � ∂ ± 1 � [ T ] M p ( k [ x p ][ ∂ ± p ][ T ])

  50. x ) with the determinant. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm proof of the commutativity Step 1 : Isomorphism with a matrix algebra afuer scalar extension. Step 2 : The determinant : restriction, corestriction Step 3 : Equility of (resp. Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . M θ M p ( k [ θ p − θ ][ ∂ ± p ][ T ]) ∼ k [ θ ] � ∂ ± 1 � [ T ] ∼ ∼ M x ∼ k [ x ] � ∂ ± 1 � [ T ] M p ( k [ x p ][ ∂ ± p ][ T ])

  51. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm proof of the commutativity Step 1 : Isomorphism with a matrix algebra afuer scalar extension. Step 2 : The determinant : restriction, corestriction Raphaël Pagès . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . M θ M p ( k [ θ p − θ ][ ∂ ± p ][ T ]) ∼ k [ θ ] � ∂ ± 1 � [ T ] ∼ ∼ M x ∼ k [ x ] � ∂ ± 1 � [ T ] M p ( k [ x p ][ ∂ ± p ][ T ]) Step 3 : Equility of Ξ θ,∂ (resp. Ξ x ,∂ ) with the determinant.

  52. p T p k x p . k x . . . . . . . Introduction Mathematical theory Algorithm Step 1 : Isomorphism with a matrix algebra k x T . k x p T T ... T p and ... p Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures T p − T = θ p − θ or T p − T = x p ∂ p

  53. . Introduction . . . . . . . . . . Mathematical theory . Algorithm Step 1 : Isomorphism with a matrix algebra T T ... T p and ... p Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures T p − T = θ p − θ or T p − T = x p ∂ p k [ x ] � ∂ ± 1 � [ T ] = k [ x ] � ∂ ± 1 � ⊗ k [ x p ][ ∂ ± p ] k [ x p ][ ∂ ± p ][ T ]

  54. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Step 1 : Isomorphism with a matrix algebra T ... ... Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . T p − T = θ p − θ or T p − T = x p ∂ p k [ x ] � ∂ ± 1 � [ T ] = k [ x ] � ∂ ± 1 � ⊗ k [ x p ][ ∂ ± p ] k [ x p ][ ∂ ± p ][ T ]     1 T + 1     M θ ( θ ) =  and M θ ( ∂ ) =         1    T + p − 1 ∂ p

  55. x T x T x T M p . . . . . . . Introduction Mathematical theory Algorithm Step 2 : The determinant, restriction, corestriction . . T M p T T x x det det Invariance by T T a Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures = k [ · ] � ∂ ± 1 � D · Z · = le centre associé

  56. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Step 2 : The determinant, restriction, corestriction det det Invariance by T T a Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . N x M x D x [ T ] M p ( Z x [ T ]) Z x [ T ] = k [ · ] � ∂ ± 1 � D · ∼ ∼ ∼ Z · = le centre associé M θ D θ [ T ] M p ( Z θ [ T ]) Z θ [ T ] N θ

  57. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Step 2 : The determinant, restriction, corestriction det det Invariance by T T a Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . N x M x D x [ T ] M p ( Z x [ T ]) Z x [ T ] = k [ · ] � ∂ ± 1 � D · ∼ ∼ ∼ Z · = le centre associé M θ D θ [ T ] M p ( Z θ [ T ]) Z θ [ T ] N θ N · ( D · ) ⊂ Z ·

  58. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Step 2 : The determinant, restriction, corestriction det det Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . N x M x D x [ T ] M p ( Z x [ T ]) Z x [ T ] = k [ · ] � ∂ ± 1 � D · ∼ ∼ ∼ Z · = le centre associé M θ D θ [ T ] M p ( Z θ [ T ]) Z θ [ T ] N θ N · ( D · ) ⊂ Z · Invariance by T �→ T + a

  59. p T p T M p k x p p T p T M p k . . . Introduction Mathematical theory L p Step 3 : Equality with the determinant Lemma . is multiplicative. L Algorithm . k x . k x p k p k p x x det det Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures N · ( L ) and Ξ · ,∂ ( L ) have the same leading coefgicient.

  60. p T p T M p k x p p T p T M p k . . . . Introduction L p Algorithm Step 3 : Equality with the determinant . Lemma L Mathematical theory . k x . k x p k p k p x x det det Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures N · ( L ) and Ξ · ,∂ ( L ) have the same leading coefgicient. N · is multiplicative.

  61. p T p T M p k x p p T p T M p k Step 3 : Equality with the determinant Algorithm Mathematical theory . Lemma . . . . . . Introduction k x p k x . k p k p x x det det Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures N · ( L ) and Ξ · ,∂ ( L ) have the same leading coefgicient. N · is multiplicative. N · ( L ∈ Z · ) = L p

  62. p T p T M p k x p p T p T M p k Step 3 : Equality with the determinant Algorithm Mathematical theory . Lemma . . . . . . Introduction k x p k x . k p k p x x det det Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures N · ( L ) and Ξ · ,∂ ( L ) have the same leading coefgicient. N · is multiplicative. N · ( L ∈ Z · ) = L p

  63. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Step 3 : Equality with the determinant Lemma det det Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . N · ( L ) and Ξ · ,∂ ( L ) have the same leading coefgicient. N · is multiplicative. N · ( L ∈ Z · ) = L p Ξ x ,∂ M x k [ x ] � ∂ ± 1 � M p ( k [ x p ][ ∂ ± p ][ T ]) k [ x p ][ ∂ ± p ][ T ] ∼ ∼ ∼ M θ k [ θ ] � ∂ ± 1 � M p ( k [ θ p ][ ∂ ± p ][ T ]) k [ θ p ][ ∂ ± p ][ T ] Ξ θ,∂

  64. p L for all p p x p x . Step 1 : Compute the B p p p p The algorithm’s skeleton p Algorithm Mathematical theory Introduction . . . p N . . x p n M n p p p B p Step 2 : Compute their characteristic polynomials degree : O p Step 3 : Compute their reciproqual image by p . Size of the output at the end of step 2 : O N Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . L ∈ Z [ x ] � ∂ �

  65. p L for all p p x . N . . . . . Introduction Mathematical theory Algorithm The algorithm’s skeleton p p Step 1 : Compute the B p p x . p n M n p p p B p Step 2 : Compute their characteristic polynomials degree : O p Step 3 : Compute their reciproqual image by p . Size of the output at the end of step 2 : O N Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � −

  66. p L for all p p x x . . . . . Introduction Mathematical theory Algorithm The algorithm’s skeleton Step 1 : Compute the B p p N . . . p n M n p p p B p Step 2 : Compute their characteristic polynomials degree : O p Step 3 : Compute their reciproqual image by p . Size of the output at the end of step 2 : O N Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p

  67. . . . . . . . . . . . . Introduction . Mathematical theory Algorithm The algorithm’s skeleton B p Step 2 : Compute their characteristic polynomials degree : O p Step 3 : Compute their reciproqual image by p . Size of the output at the end of step 2 : O N Raphaël Pagès . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p Step 1 : Compute the B p ◦ Φ p ◦ π p ( L ) for all p ⩽ N . Z [ x ] � ∂ � π p Φ p F p [ θ ] � ∂ ± 1 � F p [ x ] � ∂ � ⨿ n ∈ N M n ( F p ( θ ))

  68. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm The algorithm’s skeleton B p Step 2 : Compute their characteristic polynomials Step 3 : Compute their reciproqual image by p . Size of the output at the end of step 2 : O N Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p Step 1 : Compute the B p ◦ Φ p ◦ π p ( L ) for all p ⩽ N . Z [ x ] � ∂ � π p Φ p F p [ θ ] � ∂ ± 1 � F p [ x ] � ∂ � ⨿ n ∈ N M n ( F p ( θ )) degree : O ( p )

  69. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm The algorithm’s skeleton B p Step 2 : Compute their characteristic polynomials Size of the output at the end of step 2 : O N Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p Step 1 : Compute the B p ◦ Φ p ◦ π p ( L ) for all p ⩽ N . Z [ x ] � ∂ � π p Φ p F p [ θ ] � ∂ ± 1 � F p [ x ] � ∂ � ⨿ n ∈ N M n ( F p ( θ )) degree : O ( p ) Step 3 : Compute their reciproqual image by Φ p .

  70. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm The algorithm’s skeleton B p Step 2 : Compute their characteristic polynomials Size of the output at the end of step 2 : O N Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p Step 1 : Compute the B p ◦ Φ p ◦ π p ( L ) for all p ⩽ N . Z [ x ] � ∂ � π p Φ p F p [ θ ] � ∂ ± 1 � F p [ x ] � ∂ � ⨿ n ∈ N M n ( F p ( θ )) degree : O ( p ) Step 3 : Compute their reciproqual image by Φ p .

  71. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm The algorithm’s skeleton B p Step 2 : Compute their characteristic polynomials Raphaël Pagès . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p Step 1 : Compute the B p ◦ Φ p ◦ π p ( L ) for all p ⩽ N . Z [ x ] � ∂ � π p Φ p F p [ θ ] � ∂ ± 1 � F p [ x ] � ∂ � ⨿ n ∈ N M n ( F p ( θ )) degree : O ( p ) Step 3 : Compute their reciproqual image by Φ p . Size of the output at the end of step 2 : O ( N 2 )

  72. . p p p d P dp . deg P i . k p List of P Coefgicients of L of degree d in x . Algorithm Mathematical theory Introduction . d p . p Raphaël Pagès i p i p i p i Lemma x p p p p p dp p dp P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Step 3 : computing modulo θ d

  73. . p . . . Introduction Mathematical theory Algorithm Coefgicients of L of degree d in x . deg P i dp . P p d p d p p . P p dp dp p p p x p p Lemma i p p i i p i Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . Step 3 : computing modulo θ d List of P ∈ k [ θ p − θ ] .

  74. . p . . . . Introduction Mathematical theory Algorithm Coefgicients of L of degree d in x . P p d p d p p . P p dp dp p p p x p p Lemma i p p i i p i Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Step 3 : computing modulo θ d List of P ∈ k [ θ p − θ ] . deg ( P i ) ⩽ dp .

  75. . P . . . . . . . . Introduction Mathematical theory Algorithm Coefgicients of L of degree d in x . p dp . dp p p p x p p Lemma i p p i i p i Raphaël Pagès . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . Step 3 : computing modulo θ d List of P ∈ k [ θ p − θ ] . deg ( P i ) ⩽ dp . P = p d ( θ p − θ ) d + . . . + p 1 ( θ p − θ ) + p 0

  76. . Introduction . . . . . . . . . . Mathematical theory . Algorithm Coefgicients of L of degree d in x . p x p p Lemma i p p i i p i Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Step 3 : computing modulo θ d List of P ∈ k [ θ p − θ ] . deg ( P i ) ⩽ dp . P = p d ( θ p − θ ) d + . . . + p 1 ( θ p − θ ) + p 0 dp θ dp + . . . + p ′ P = p ′ 1 θ + p 0

  77. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Coefgicients of L of degree d in x . Lemma i p p i i p i Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . Step 3 : computing modulo θ d List of P ∈ k [ θ p − θ ] . deg ( P i ) ⩽ dp . P = p d ( θ p − θ ) d + . . . + p 1 ( θ p − θ ) + p 0 dp θ dp + . . . + p ′ P = p ′ 1 θ + p 0 θ p − θ �→ x p ∂ p

  78. . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Coefgicients of L of degree d in x . Lemma i Raphaël Pagès . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . Step 3 : computing modulo θ d List of P ∈ k [ θ p − θ ] . deg ( P i ) ⩽ dp . P = p d ( θ p − θ ) d + . . . + p 1 ( θ p − θ ) + p 0 dp θ dp + . . . + p ′ P = p ′ 1 θ + p 0 θ p − θ �→ x p ∂ p ∀ i ⩽ p − 1 p i = ( − 1) i p ′

  79. . . . . . . . . . . . . Introduction . Mathematical theory Algorithm Structure of the algorithm Step 2 : Compute their characteristic polynomials mod d . O N . Step 3 : Compute their reciproqual image by p . O N Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p Step 1 : Compute the B p ◦ Φ p ◦ π p ( L ) mod θ d for all p ⩽ N .

  80. . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Structure of the algorithm Step 2 : Compute their characteristic polynomials Step 3 : Compute their reciproqual image by p . O N Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p Step 1 : Compute the B p ◦ Φ p ◦ π p ( L ) mod θ d for all p ⩽ N . mod θ d . ˜ O ( N ) .

  81. . . . . . . . . . . . . . . . . Introduction Mathematical theory Algorithm Structure of the algorithm Step 2 : Compute their characteristic polynomials Raphaël Pagès . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . ∼ Φ p : F p [ x ] � ∂ ± 1 � → F p [ θ ] � ∂ ± 1 � L ∈ Z [ x ] � ∂ � − π p : Z → F p Step 1 : Compute the B p ◦ Φ p ◦ π p ( L ) mod θ d for all p ⩽ N . mod θ d . ˜ O ( N ) . Step 3 : Compute their reciproqual image by Φ p . ˜ O ( N )

  82. . Algorithm . . . . . . . . . Introduction Mathematical theory HaRvey, 2014] . mod s s s mod s s mod s mod s s s mod s s Raphaël Pagès . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . Computation of ( p − 1)! mod p s [Costa, GeRbicz, N = 7 . (3 − 1)!

  83. . Algorithm . . . . . . . . . Introduction Mathematical theory HaRvey, 2014] . mod s s s mod s s mod s mod s s s mod s s Raphaël Pagès . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efgicient computation of p -curvatures Computation of ( p − 1)! mod p s [Costa, GeRbicz, N = 7 . (3 − 1)! (5 − 1)! (7 − 1)!

  84. . Introduction . . . . . . . . . . Mathematical theory . Algorithm HaRvey, 2014] mod s s mod s mod s s s mod s s Raphaël Pagès . . . . . . . . . . . . . . . Efgicient computation of p -curvatures . . . . . . . . . . . . . Computation of ( p − 1)! mod p s [Costa, GeRbicz, N = 7 . (3 − 1)! (5 − 1)! (7 − 1)! mod 3 s 5 s 7 s

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