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Prcision p -adique Prcision p -adique Journes Nationales du Calcul Formel 2014 X.Caruso, P.Lairez, D.Roe, T.Vaccon University of Rennes I, TU Berlin, University of Calgary, University of Rennes I 6 novembre 2014 X.Caruso, P.Lairez, D.Roe,

  1. Précision p -adique p -adic precision p -adic algorithms and step-by-step analysis Optimality Step-by-step analysis is not optimal. Q 2 Q 2 → p p Let f : ( x , y ) �→ ( x + y , x − y ) . We would like to compute f ◦ f ( x , y ) with ( x , y ) = (1 + O ( p 10 ) , 1 + O ( p )) . If we apply f two times, we get : f ◦ f ( x , y ) = (2 + O ( p ) , 2 + O ( p )) .

  2. Précision p -adique p -adic precision p -adic algorithms and step-by-step analysis Optimality Step-by-step analysis is not optimal. Q 2 Q 2 → p p Let f : ( x , y ) �→ ( x + y , x − y ) . We would like to compute f ◦ f ( x , y ) with ( x , y ) = (1 + O ( p 10 ) , 1 + O ( p )) . If we apply f two times, we get : f ◦ f ( x , y ) = (2 + O ( p ) , 2 + O ( p )) . If we remark that f ◦ f = 2 Id , we get : f ◦ f ( x , y ) = (2 + O ( p 10 ) , 2 + O ( p )) . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  3. Précision p -adique p -adic precision p -adic algorithms and step-by-step analysis Optimality Step-by-step analysis is not optimal. Q 2 Q 2 → p p Let f : ( x , y ) �→ ( x + y , x − y ) . We would like to compute f ◦ f ( x , y ) with ( x , y ) = (1 + O ( p 10 ) , 1 + O ( p )) . If we apply f two times, we get : f ◦ f ( x , y ) = (2 + O ( p ) , 2 + O ( p )) . If we remark that f ◦ f = 2 Id , we get : f ◦ f ( x , y ) = (2 + O ( p 10 ) , 2 + O ( p )) . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  4. Précision p -adique p -adic precision The differential approach Table of contents 1 p -adic precision p -adic algorithms and step-by-step analysis The differential approach Improvements 2 Precision in practice Linear Differential Equation (cf BGPS) About implementation Lifting method X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  5. Précision p -adique p -adic precision The differential approach The Main lemma of p -adic differential precision Lemma Let f : Q n p → Q m p be a differentiable mapping.

  6. Précision p -adique p -adic precision The differential approach The Main lemma of p -adic differential precision Lemma Let f : Q n p → Q m p be a differentiable mapping. Let x ∈ Q n p . We assume that f ′ ( x ) is surjective .

  7. Précision p -adique p -adic precision The differential approach The Main lemma of p -adic differential precision Lemma Let f : Q n p → Q m p be a differentiable mapping. Let x ∈ Q n p . We assume that f ′ ( x ) is surjective . Then for any ball B = B (0 , r ) small enough , f ( x + B ) = f ( x ) + f ′ ( x ) · B . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  8. Précision p -adique p -adic precision The differential approach The Main lemma of p -adic differential precision Lemma Let f : Q n p → Q m p be a differentiable mapping. Let x ∈ Q n p . We assume that f ′ ( x ) is surjective . Then for any ball B = B (0 , r ) small enough , f ( x + B ) = f ( x ) + f ′ ( x ) · B . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  9. Précision p -adique p -adic precision The differential approach Geometrical meaning Interpretation x f ( x ) B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  10. Précision p -adique p -adic precision The differential approach Geometrical meaning Interpretation x f ( x ) f ′ ( x ) B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  11. Précision p -adique p -adic precision The differential approach Geometrical meaning Interpretation x f ( x ) f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  12. Précision p -adique p -adic precision The differential approach Geometrical meaning Interpretation x f ( x ) x + B f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  13. Précision p -adique p -adic precision The differential approach Geometrical meaning Interpretation x f ( x ) x + B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  14. Précision p -adique p -adic precision The differential approach Geometrical meaning Interpretation x f ( x ) x + B f ( x ) + f ′ ( x ) · B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  15. Précision p -adique p -adic precision Improvements Table of contents 1 p -adic precision p -adic algorithms and step-by-step analysis The differential approach Improvements 2 Precision in practice Linear Differential Equation (cf BGPS) About implementation Lifting method X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  16. Précision p -adique p -adic precision Improvements Lattices X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  17. Précision p -adique p -adic precision Improvements Lattices Lemma Let f : Q n p → Q m p be a differentiable mapping. Let x ∈ Q n p . We assume that f ′ ( x ) is surjective . Then for any ball B = B (0 , r ) small enough , f ( x + B ) = f ( x ) + f ′ ( x ) · B . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  18. Précision p -adique p -adic precision Improvements Lattices Lemma Let f : Q n p → Q m p be a differentiable mapping. Let x ∈ Q n p . We assume that f ′ ( x ) is surjective . Then for any ball B = B (0 , r ) small enough , for any open lattice H ⊂ B f ( x + H ) = f ( x ) + f ′ ( x ) · H . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  19. Précision p -adique p -adic precision Improvements Lattices Lemma Let f : Q n p → Q m p be a differentiable mapping. Let x ∈ Q n p . We assume that f ′ ( x ) is surjective . Then for any ball B = B (0 , r ) small enough , for any open lattice H ⊂ B f ( x + H ) = f ( x ) + f ′ ( x ) · H . Remark This allows more models of precision, like ( x , y ) = (1 + O ( p 10 ) , 1 + O ( p )) . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  20. Précision p -adique p -adic precision Improvements Lattices Lemma Let f : Q n p → Q m p be a differentiable mapping. Let x ∈ Q n p . We assume that f ′ ( x ) is surjective . Then for any ball B = B (0 , r ) small enough , for any open lattice H ⊂ B f ( x + H ) = f ( x ) + f ′ ( x ) · H . Remark This allows more models of precision, like ( x , y ) = (1 + O ( p 10 ) , 1 + O ( p )) . Remark Our framework can be extended to (complete) ultrametric K -vector spaces ( e.g. F p (( X )) n , Q (( X )) m ). X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  21. Précision p -adique p -adic precision Improvements Higher differentials X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  22. Précision p -adique p -adic precision Improvements Higher differentials What is small enough ? How can we determine when the lemma applies ? X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  23. Précision p -adique p -adic precision Improvements Higher differentials What is small enough ? How can we determine when the lemma applies ? When f is locally analytic, it essentially corresponds to + ∞ � 1 k ! f ( k ) ( x ) · H k ⊂ f ′ ( x ) · H . k =2 X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  24. Précision p -adique p -adic precision Improvements Higher differentials What is small enough ? How can we determine when the lemma applies ? When f is locally analytic, it essentially corresponds to + ∞ � 1 k ! f ( k ) ( x ) · H k ⊂ f ′ ( x ) · H . k =2 This can be determined with Newton-polygon techniques. X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  25. Précision p -adique p -adic precision Improvements Some differentiable operations Some more examples We can apply our method to: X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  26. Précision p -adique p -adic precision Improvements Some differentiable operations Some more examples We can apply our method to: On matrices: determinant, characteristic polynomial, LU factorization... X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  27. Précision p -adique p -adic precision Improvements Some differentiable operations Some more examples We can apply our method to: On matrices: determinant, characteristic polynomial, LU factorization... On Q p [ X ] : evaluation, interpolation, GCD, factorization... X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  28. Précision p -adique p -adic precision Improvements Some differentiable operations Some more examples We can apply our method to: On matrices: determinant, characteristic polynomial, LU factorization... On Q p [ X ] : evaluation, interpolation, GCD, factorization... On Q p [ X 1 , . . . , X n ] : division, Gröbner bases. X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  29. Précision p -adique p -adic precision Improvements Some differentiable operations Some more examples We can apply our method to: On matrices: determinant, characteristic polynomial, LU factorization... On Q p [ X ] : evaluation, interpolation, GCD, factorization... On Q p [ X 1 , . . . , X n ] : division, Gröbner bases. X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  30. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Table of contents 1 p -adic precision p -adic algorithms and step-by-step analysis The differential approach Improvements 2 Precision in practice Linear Differential Equation (cf BGPS) About implementation Lifting method X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  31. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Newton series Definition Let P ∈ k [ X ], P = � i ( X − α i ) (over k ). The Newton series of P is � � i ) t n = − ( P ∗ ) ′ α n f P = ( P ∗ , n i with P ∗ the reciprocal polynomial of P . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  32. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Newton series Definition Let P ∈ k [ X ], P = � i ( X − α i ) (over k ). The Newton series of P is � � i ) t n = − ( P ∗ ) ′ α n f P = ( P ∗ , n i with P ∗ the reciprocal polynomial of P . Remark When char ( k ) = 0 , by Newton’s identities, we can recover P from f P . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  33. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation over Newton series Proposition Let P × Q = � i ( X − α i ) � j ( X − β j ) and P ⊗ Q = � i , j ( X − α i β j ) . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  34. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation over Newton series Proposition Let P × Q = � i ( X − α i ) � j ( X − β j ) and P ⊗ Q = � i , j ( X − α i β j ) . Then f P × Q = f P + f Q X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  35. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation over Newton series Proposition Let P × Q = � i ( X − α i ) � j ( X − β j ) and P ⊗ Q = � i , j ( X − α i β j ) . Then f P × Q = f P + f Q and f P ⊗ Q is obtained by the product coefficient by coefficient of f P and f Q . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  36. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation over Newton series Proposition Let P × Q = � i ( X − α i ) � j ( X − β j ) and P ⊗ Q = � i , j ( X − α i β j ) . Then f P × Q = f P + f Q and f P ⊗ Q is obtained by the product coefficient by coefficient of f P and f Q . Remark How to do that on Z / p Z ? X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  37. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation over Newton series Proposition Let P × Q = � i ( X − α i ) � j ( X − β j ) and P ⊗ Q = � i , j ( X − α i β j ) . Then f P × Q = f P + f Q and f P ⊗ Q is obtained by the product coefficient by coefficient of f P and f Q . Remark How to do that on Z / p Z ? We can lift Z / p Z to Z p , and reduce mod p afterwards. X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  38. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Idea of the algorithm Proposition (How to recover P with f P ) We notice that ( P ∗ ) ′ = − f P ∗ P ∗ , X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  39. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Idea of the algorithm Proposition (How to recover P with f P ) We notice that ( P ∗ ) ′ = − f P ∗ P ∗ , Therefore y ′ = a ( x ) ∗ y , with a ( x ) = − f P and y = P ∗ . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  40. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Idea of the algorithm Proposition (How to recover P with f P ) We notice that ( P ∗ ) ′ = − f P ∗ P ∗ , Therefore y ′ = a ( x ) ∗ y , with a ( x ) = − f P and y = P ∗ . Thus, we can recover P with f P by solving a linear differential equation (in Z p [[ x ]] ). X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  41. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  42. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. We differentiate the mapping sending a to the solution of y ′ = a ( t ) y . y 0 corresponds to P ∗ , and a to − f P , in Z p [[ x ]] X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  43. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. We differentiate the mapping sending a to the solution of y ′ = a ( t ) y . ( y 0 + δ y ) ′ = ( a + δ a )( y 0 + δ y ) X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  44. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. We differentiate the mapping sending a to the solution of y ′ = a ( t ) y . ( y 0 + δ y ) ′ = ( a + δ a )( y 0 + δ y ) ( δ y ) ′ = a δ y + y 0 δ a X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  45. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. We differentiate the mapping sending a to the solution of y ′ = a ( t ) y . ( δ y ) ′ = a δ y + y 0 δ a X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  46. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. We differentiate the mapping sending a to the solution of y ′ = a ( t ) y . ( δ y ) ′ = a δ y + y 0 δ a By variation of parameters, � y − 1 δ y = y 0 × y 0 δ a d t . 0 X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  47. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. By variation of parameters, � y − 1 δ y = y 0 × y 0 δ a d t . 0 X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  48. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. By variation of parameters, � y − 1 δ y = y 0 × y 0 δ a d t . 0 � δ y = y 0 δ a d t . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  49. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. � δ y = y 0 δ a d t . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  50. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. � δ y = y 0 δ a d t . � � i a i x i = � i a i x i +1 Since i +1 , it yields a logarithmic loss in precision. X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  51. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. � δ y = y 0 δ a d t . In other words, if δ a = O ( p k ), i.e. δ a = � i u i p k t i , then δ y = � p k i +1 t i . i v i X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  52. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Computation of the differential Theorem The loss in precision to recover the n-th coefficient of P with f P is in O ( log p ( n ) ) . Proof. � δ y = y 0 δ a d t . In other words, if δ a = O ( p k ), i.e. δ a = � i u i p k t i , then δ y = � p k i +1 t i . i v i X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  53. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Newton polygons technique Newton and Legendre X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  54. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Newton polygons technique Newton and Legendre log � � 2 k order Newton polygon of f ≥ 2 , − 2 m convex lower-bound of ( n , � f n � ) n ≥ 2 − km X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  55. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Newton polygons technique Newton and Legendre log ε − 2 m − m log ε − 2 m + 2 log ε NP ( f ≥ 2 ) ∗ y = 2 x + 2 m X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  56. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Newton polygons technique Newton and Legendre log ε − 2 m − m y = x + log ε log ε − 2 m + 2 log ε NP ( f ≥ 2 ) ∗ y = 2 x + 2 m X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  57. Précision p -adique Precision in practice Linear Differential Equation (cf BGPS) Newton polygons technique Newton and Legendre log ε − 2 m − m y = x + log ε log ε − 2 m + 2 log ε NP ( f ≥ 2 ) ∗ y = 2 x + 2 m X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  58. Précision p -adique Precision in practice About implementation Table of contents 1 p -adic precision p -adic algorithms and step-by-step analysis The differential approach Improvements 2 Precision in practice Linear Differential Equation (cf BGPS) About implementation Lifting method X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  59. Précision p -adique Precision in practice About implementation Newton scheme Proposition One can compute y 0 through some Newton scheme : � u l +1 ← u l + u ′ l ) a + O ( x 2 l +1 +1 ) . l ( a × u l − u ′ u ′ a l X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  60. Précision p -adique Precision in practice About implementation Newton scheme Proposition One can compute y 0 through some Newton scheme : � u l +1 ← u l + u ′ l ) a + O ( x 2 l +1 +1 ) . l ( a × u l − u ′ u ′ a l Remark � O ( p N ) x k = O ( p N ) k + 1 x k +1 . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  61. Précision p -adique Precision in practice About implementation Newton scheme Proposition One can compute y 0 through some Newton scheme : � u l +1 ← u l + u ′ l ) a + O ( x 2 l +1 +1 ) . l ( a × u l − u ′ u ′ a l Remark � O ( p N ) x k = O ( p N ) k + 1 x k +1 . One lose E (log p (2 l )) at each step. There are E (log 2 (2 N )) steps to reach x 2 N , it means a naïve loss in precision to compute the coefficient of x 2 N � (log 2 N ) 2 � in . X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  62. Précision p -adique Precision in practice About implementation What happens in practice ? Figure: Precision over the output X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  63. Précision p -adique Precision in practice About implementation What happens in practice ? Figure: Precision over the output X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  64. Précision p -adique Precision in practice About implementation What happens in practice ? Figure: Precision over the output X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  65. Précision p -adique Precision in practice About implementation What happens in practice ? Figure: Precision over the output X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  66. Précision p -adique Precision in practice Lifting method Table of contents 1 p -adic precision p -adic algorithms and step-by-step analysis The differential approach Improvements 2 Precision in practice Linear Differential Equation (cf BGPS) About implementation Lifting method X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  67. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N ) x B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  68. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N ) x f ′ ( x ) B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  69. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N ) x f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  70. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N ) ? + O ( p N ) x x + B f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  71. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N ) ? + O ( p N ) x x + B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  72. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N ) ? + O ( p N ) x x + B f ( x ) ? f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  73. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision � x + O ( p N ) f ( x ) + O ( p M ) � f x x + B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  74. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision � x + O ( p N ) f ( x ) + O ( p M ) � f x x + B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  75. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N + N ′ ) � f x x + B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  76. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N + N ′ ) y + O ( p M ′ ) � f x x + B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  77. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N + N ′ ) y + O ( p M ′ ) ⊂ f ( x ) + O ( p N ) � f x x + B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  78. Précision p -adique Precision in practice Lifting method More on the differential method Differential tracking of precision x + O ( p N + N ′ ) y + O ( p M ′ ) ⊂ f ( x ) + O ( p N ) � f x x + B f ( x ) + f ′ ( x ) · B f f ′ ( x ) f ′ ( x ) · B B X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

  79. Précision p -adique Precision in practice Lifting method Buffer method Proposition One can use the following Newton scheme with lift: Lift u l to O ( p N + m ) , a + O ( x 2 l ) ← u l × u ′− 1 + O ( p N + m , x 2 l ) . l � u l +1 ← u l + u ′ l + O ( x 2 l +1 +1 ) . l ) a ( a × u l − u ′ l u ′ a X.Caruso, P.Lairez, D.Roe, T.Vaccon Précision p -adique

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