Division and Slope Factorization of P-Adic Polynomials

Division and Slope Factorization of p-Adic Polynomials

Xavier Caruso, David Roe Tristan Vaccon

Univ.Rennes 1, Univ. Pittsburgh, 立教大学

July 22nd, 2016

...... Division and Slope Factorization of p-Adic Polynomials Introduction

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... e.g. Dixon’s method (used in F4), Polynomial factorization via Hensel’s lemma.

p-adic algorithms

Going from Z/pZ to Zp and then back to Z/pZ enables more computation, e.g. the algorithms of Bostan et al. and Lercier et al. using p-adic diﬀerential equations ; Kedlaya’s and Lauder’s counting-point algorithms via p-adic cohomology ;

My personal (long-term) motivation Computing (some) moduli spaces of p-adic Galois representations.

Division and Slope Factorization of p-Adic Polynomials Introduction

Why should one work with p-adic numbers ?

p-adic methods

Working in Qp instead of Q, one can handle more eﬃciently the coeﬃcients growth ;

...... p-adic algorithms

My personal (long-term) motivation Computing (some) moduli spaces of p-adic Galois representations.

Division and Slope Factorization of p-Adic Polynomials Introduction

Why should one work with p-adic numbers ?

p-adic methods

Working in Qp instead of Q, one can handle more eﬃciently the coeﬃcients growth ; e.g. Dixon’s method (used in F4), Polynomial factorization via Hensel’s lemma.

...... e.g. the algorithms of Bostan et al. and Lercier et al. using p-adic diﬀerential equations ; Kedlaya’s and Lauder’s counting-point algorithms via p-adic cohomology ;

My personal (long-term) motivation Computing (some) moduli spaces of p-adic Galois representations.

Division and Slope Factorization of p-Adic Polynomials Introduction

Why should one work with p-adic numbers ?

p-adic methods

Working in Qp instead of Q, one can handle more eﬃciently the coeﬃcients growth ; e.g. Dixon’s method (used in F4), Polynomial factorization via Hensel’s lemma.

p-adic algorithms

Going from Z/pZ to Zp and then back to Z/pZ enables more computation,

...... Kedlaya’s and Lauder’s counting-point algorithms via p-adic cohomology ;

My personal (long-term) motivation Computing (some) moduli spaces of p-adic Galois representations.

Division and Slope Factorization of p-Adic Polynomials Introduction

Why should one work with p-adic numbers ?

p-adic methods

Working in Qp instead of Q, one can handle more eﬃciently the coeﬃcients growth ; e.g. Dixon’s method (used in F4), Polynomial factorization via Hensel’s lemma.

p-adic algorithms

Going from Z/pZ to Zp and then back to Z/pZ enables more computation, e.g. the algorithms of Bostan et al. and Lercier et al. using p-adic diﬀerential equations ;

...... My personal (long-term) motivation Computing (some) moduli spaces of p-adic Galois representations.

Division and Slope Factorization of p-Adic Polynomials Introduction

Why should one work with p-adic numbers ?

p-adic methods

Working in Qp instead of Q, one can handle more eﬃciently the coeﬃcients growth ; e.g. Dixon’s method (used in F4), Polynomial factorization via Hensel’s lemma.

p-adic algorithms

...... Division and Slope Factorization of p-Adic Polynomials Introduction

Why should one work with p-adic numbers ?

p-adic methods

Working in Qp instead of Q, one can handle more eﬃciently the coeﬃcients growth ; e.g. Dixon’s method (used in F4), Polynomial factorization via Hensel’s lemma.

p-adic algorithms

My personal (long-term) motivation

Computing (some) moduli spaces of p-adic Galois. representations...... This year, we study basic operations related to polynomial computations.

More motivations Understanding basic operations related to ﬁeld extensions, in particular division and quotients. Understanding the behaviour of precision during factorisation: over Qp or kJT K, or as an intermediate to factorisation over Q.

Today’s highlights Optimal tracking of precision for modular multiplication and diﬀused digits. Precision for slope factorization algorithms.

Division and Slope Factorization of p-Adic Polynomials Introduction

Studying polynomial computations over p-adics

A building block At ISSAC 2015, we have studied the p-adic stability of some computations in linear algebra.

...... More motivations Understanding basic operations related to ﬁeld extensions, in particular division and quotients. Understanding the behaviour of precision during factorisation: over Qp or kJT K, or as an intermediate to factorisation over Q.

Today’s highlights Optimal tracking of precision for modular multiplication and diﬀused digits. Precision for slope factorization algorithms.

Division and Slope Factorization of p-Adic Polynomials Introduction

Studying polynomial computations over p-adics

A building block At ISSAC 2015, we have studied the p-adic stability of some computations in linear algebra. This year, we study basic operations related to polynomial computations.

...... Understanding the behaviour of precision during factorisation: over Qp or kJT K, or as an intermediate to factorisation over Q.

Today’s highlights Optimal tracking of precision for modular multiplication and diﬀused digits. Precision for slope factorization algorithms.

Division and Slope Factorization of p-Adic Polynomials Introduction

Studying polynomial computations over p-adics

A building block At ISSAC 2015, we have studied the p-adic stability of some computations in linear algebra. This year, we study basic operations related to polynomial computations.

More motivations Understanding basic operations related to ﬁeld extensions, in particular division and quotients.

...... Today’s highlights Optimal tracking of precision for modular multiplication and diﬀused digits. Precision for slope factorization algorithms.

Division and Slope Factorization of p-Adic Polynomials Introduction

Studying polynomial computations over p-adics

A building block At ISSAC 2015, we have studied the p-adic stability of some computations in linear algebra. This year, we study basic operations related to polynomial computations.

...... Precision for slope factorization algorithms.

Division and Slope Factorization of p-Adic Polynomials Introduction

Studying polynomial computations over p-adics

A building block At ISSAC 2015, we have studied the p-adic stability of some computations in linear algebra. This year, we study basic operations related to polynomial computations.

Today’s highlights Optimal tracking of precision for modular multiplication and diﬀused digits.

...... Division and Slope Factorization of p-Adic Polynomials Introduction

Studying polynomial computations over p-adics

A building block At ISSAC 2015, we have studied the p-adic stability of some computations in linear algebra. This year, we study basic operations related to polynomial computations.

Today’s highlights Optimal tracking of precision for modular multiplication and diﬀused digits. Precision for slope factorization algorithms...... p refers to a prime number

p-adic numbers are numbers written in p-basis of the shape:

a = ... ai ... a2 a1 a0 , a−1 a−2 ... a−n

with 0 ≤ ai < p for all i. Addition and multiplication on these numbers are deﬁned by applying SchoolBook algorithms.

The valuation vp(a) of a is the smallest v such that av ≠ 0.

The p-adic numbers form the ﬁeld Qp.

A p-adic number with no digit after the comma is a p-adic integer.

The p-adic integers form a subring Zp of Qp.

Division and Slope Factorization of p-Adic Polynomials Introduction

What are p-adic numbers?

...... p-adic numbers are numbers written in p-basis of the shape:

a = ... ai ... a2 a1 a0 , a−1 a−2 ... a−n

with 0 ≤ ai < p for all i. Addition and multiplication on these numbers are deﬁned by applying SchoolBook algorithms.

The valuation vp(a) of a is the smallest v such that av ≠ 0.

The p-adic numbers form the ﬁeld Qp.

A p-adic number with no digit after the comma is a p-adic integer.

The p-adic integers form a subring Zp of Qp.

Division and Slope Factorization of p-Adic Polynomials Introduction

What are p-adic numbers?

p refers to a prime number

...... Addition and multiplication on these numbers are deﬁned by applying SchoolBook algorithms.

The valuation vp(a) of a is the smallest v such that av ≠ 0.

The p-adic numbers form the ﬁeld Qp.

A p-adic number with no digit after the comma is a p-adic integer.

The p-adic integers form a subring Zp of Qp.

Division and Slope Factorization of p-Adic Polynomials Introduction

What are p-adic numbers?

p refers to a prime number

p-adic numbers are numbers written in p-basis of the shape:

a = ... ai ... a2 a1 a0 , a−1 a−2 ... a−n

with 0 ≤ ai < p for all i.

...... The valuation vp(a) of a is the smallest v such that av ≠ 0.

The p-adic numbers form the ﬁeld Qp.

A p-adic number with no digit after the comma is a p-adic integer.

The p-adic integers form a subring Zp of Qp.

Division and Slope Factorization of p-Adic Polynomials Introduction

What are p-adic numbers?

p refers to a prime number

p-adic numbers are numbers written in p-basis of the shape:

a = ... ai ... a2 a1 a0 , a−1 a−2 ... a−n

with 0 ≤ ai < p for all i. Addition and multiplication on these numbers are deﬁned by applying SchoolBook algorithms.

...... The p-adic numbers form the ﬁeld Qp.

A p-adic number with no digit after the comma is a p-adic integer.

The p-adic integers form a subring Zp of Qp.

Division and Slope Factorization of p-Adic Polynomials Introduction

What are p-adic numbers?

p refers to a prime number

p-adic numbers are numbers written in p-basis of the shape:

a = ... ai ... a2 a1 a0 , a−1 a−2 ... a−n

with 0 ≤ ai < p for all i. Addition and multiplication on these numbers are deﬁned by applying SchoolBook algorithms.

The valuation vp(a) of a is the smallest v such that av ≠ 0.

...... A p-adic number with no digit after the comma is a p-adic integer.

The p-adic integers form a subring Zp of Qp.

Division and Slope Factorization of p-Adic Polynomials Introduction

What are p-adic numbers?

p refers to a prime number

p-adic numbers are numbers written in p-basis of the shape:

a = ... ai ... a2 a1 a0 , a−1 a−2 ... a−n

with 0 ≤ ai < p for all i. Addition and multiplication on these numbers are deﬁned by applying SchoolBook algorithms.

The valuation vp(a) of a is the smallest v such that av ≠ 0.

The p-adic numbers form the ﬁeld Qp.

...... The p-adic integers form a subring Zp of Qp.

Division and Slope Factorization of p-Adic Polynomials Introduction

What are p-adic numbers?

p refers to a prime number

p-adic numbers are numbers written in p-basis of the shape:

a = ... ai ... a2 a1 a0 , a−1 a−2 ... a−n

with 0 ≤ ai < p for all i. Addition and multiplication on these numbers are deﬁned by applying SchoolBook algorithms.

The valuation vp(a) of a is the smallest v such that av ≠ 0.

The p-adic numbers form the ﬁeld Qp.

A p-adic number with no digit after the comma is a p-adic integer.

...... Division and Slope Factorization of p-Adic Polynomials Introduction

What are p-adic numbers?

p refers to a prime number

p-adic numbers are numbers written in p-basis of the shape:

a = ... ai ... a2 a1 a0 , a−1 a−2 ... a−n

with 0 ≤ ai < p for all i. Addition and multiplication on these numbers are deﬁned by applying SchoolBook algorithms.

The valuation vp(a) of a is the smallest v such that av ≠ 0.

The p-adic numbers form the ﬁeld Qp.

A p-adic number with no digit after the comma is a p-adic integer.

The p-adic integers form a subring Zp of Qp.

...... k k ∀k ∈ N, Zp/p Zp = Z/p Z.

A ﬁrst idea

Qp is an extension of Q where one can perform calculus, as simply as over R. We are closer to arithmetic : we can reduce modulo p.

Remark

Qp R Zp Zp

Q Z/pZ Z/pZ

Division and Slope Factorization of p-Adic Polynomials Introduction

Summary on p-adics

Proposition

Zp/pZp = Z/pZ.

...... A ﬁrst idea

Qp is an extension of Q where one can perform calculus, as simply as over R. We are closer to arithmetic : we can reduce modulo p.

Remark

Qp R Zp Zp

Q Z/pZ Z/pZ

Division and Slope Factorization of p-Adic Polynomials Introduction

Summary on p-adics

Proposition

Zp/pZp = Z/pZ.

k k ∀k ∈ N, Zp/p Zp = Z/p Z.

...... We are closer to arithmetic : we can reduce modulo p.

Remark

Qp R Zp Zp

Q Z/pZ Z/pZ

Division and Slope Factorization of p-Adic Polynomials Introduction

Summary on p-adics

Proposition

Zp/pZp = Z/pZ.

k k ∀k ∈ N, Zp/p Zp = Z/p Z.

A ﬁrst idea

Qp is an extension of Q where one can perform calculus, as simply as over R.

...... Remark

Qp R Zp Zp

Q Z/pZ Z/pZ

Division and Slope Factorization of p-Adic Polynomials Introduction

Summary on p-adics

Proposition

Zp/pZp = Z/pZ.

k k ∀k ∈ N, Zp/p Zp = Z/p Z.

A ﬁrst idea

Qp is an extension of Q where one can perform calculus, as simply as over R. We are closer to arithmetic : we can reduce modulo p.

...... Division and Slope Factorization of p-Adic Polynomials Introduction

Summary on p-adics

Proposition

Zp/pZp = Z/pZ.

k k ∀k ∈ N, Zp/p Zp = Z/p Z.

A ﬁrst idea

Qp is an extension of Q where one can perform calculus, as simply as over R. We are closer to arithmetic : we can reduce modulo p.

Remark

Qp R Zp Zp

Q Z/pZ. . . Z. /. p. Z...... Deﬁnition ∑ d−1 i d The order, or the absolute precision of i=k ai p + O(p ) is d.

Example The order of 3 ∗ 7−1 + 4 ∗ 70 + 5 ∗ 71 + 6 ∗ 72 + O(73) is 3.

Division and Slope Factorization of p-Adic Polynomials Introduction

Deﬁnition of the precision

Finite-precision p-adics ∑ Q +∞ i ∈ J − K ∈ Z Elements of p can be written i=k ai p , with ai 0, p 1 , k and p a prime number. While working with a computer, we usually only can consider the beginning of this power serie expansion: we only consider elements of the ∑ d−1 i d ∈ Z following form i=l ai p + O(p ) , with l .

...... Deﬁnition ∑ d−1 i d The order, or the absolute precision of i=k ai p + O(p ) is d.

Example The order of 3 ∗ 7−1 + 4 ∗ 70 + 5 ∗ 71 + 6 ∗ 72 + O(73) is 3.

Division and Slope Factorization of p-Adic Polynomials Introduction

Deﬁnition of the precision

...... Example The order of 3 ∗ 7−1 + 4 ∗ 70 + 5 ∗ 71 + 6 ∗ 72 + O(73) is 3.

Division and Slope Factorization of p-Adic Polynomials Introduction

Deﬁnition of the precision

Deﬁnition ∑ d−1 i d The order, or the absolute precision of i=k ai p + O(p ) is d.

...... Division and Slope Factorization of p-Adic Polynomials Introduction

Deﬁnition of the precision

Deﬁnition ∑ d−1 i d The order, or the absolute precision of i=k ai p + O(p ) is d.

Example The order of 3 ∗ 7−1 + 4 ∗ 70 + 5 ∗ 71 + 6 ∗ 72 + O(73) is 3.

...... Remark It is quite the opposite to when dealing with real numbers, because of Round-oﬀ error : (1 + 5 ∗ 10−2) + (2 + 6 ∗ 10−2) = 3 + 1 ∗ 10−1 + 1 ∗ 10−2.

That is to say, if a and b are known up to precision 10−n, then a + b is (−n +1 ) known up to 10 .

Division and Slope Factorization of p-Adic Polynomials Introduction p-adic precion vs real precision

The quintessential idea of the step-by-step analysis is the following :

Proposition (p-adic errors don’t add) Indeed, (a + O(p k )) + (b + O(p k )) = a + b + O(p k ).

That is to say, if a and b are known up to precision O(pk ), then so is a + b.

...... Remark It is quite the opposite to when dealing with real numbers, because of Round-oﬀ error : (1 + 5 ∗ 10−2) + (2 + 6 ∗ 10−2) = 3 + 1 ∗ 10−1 + 1 ∗ 10−2.

That is to say, if a and b are known up to precision 10−n, then a + b is (−n +1 ) known up to 10 .

Division and Slope Factorization of p-Adic Polynomials Introduction p-adic precion vs real precision

The quintessential idea of the step-by-step analysis is the following :

Proposition (p-adic errors don’t add) Indeed, (a + O(p k )) + (b + O(p k )) = a + b + O(p k ).

That is to say, if a and b are known up to precision O(pk ), then so is a + b.

...... Division and Slope Factorization of p-Adic Polynomials Introduction p-adic precion vs real precision

The quintessential idea of the step-by-step analysis is the following :

Proposition (p-adic errors don’t add) Indeed, (a + O(p k )) + (b + O(p k )) = a + b + O(p k ).

That is to say, if a and b are known up to precision O(pk ), then so is a + b.

Remark It is quite the opposite to when dealing with real numbers, because of Round-oﬀ error : (1 + 5 ∗ 10−2) + (2 + 6 ∗ 10−2) = 3 + 1 ∗ 10−1 + 1 ∗ 10−2.

That is to say, if a and b are known up to precision 10−n, then a + b is (−n +1 ) known up to 10 ...... Division and Slope Factorization of p-Adic Polynomials Introduction p-adic precion vs real precision

The quintessential idea of the step-by-step analysis is the following :

Proposition (p-adic errors don’t add) Indeed, (a + O(p k )) + (b + O(p k )) = a + b + O(p k ).

That is to say, if a and b are known up to precision O(pk ), then so is a + b.

Remark It is quite the opposite to when dealing with real numbers, because of Round-oﬀ error : (1 + 5 ∗ 10−2) + (2 + 6 ∗ 10−2) = 3 + 1 ∗ 10−1 + 1 ∗ 10−2.

That is to say, if a and b are known up to precision 10−n, then a + b is (−n +1 ) known up to 10 ...... Proposition (multiplication)

k0 k1 min(k0+vp (x1),k1+vp (x0)) (x0 + O(p )) ∗ (x1 + O(p )) = x0 ∗ x1 + O(p )

Proposition (division)

xpa + O(pb) = x ∗ y −1pa−c + O(pmin(d+a−2c,b−c)) ypc + O(pd ) In particular, 1 = y −1p−c + O(pd−2c ) pc y + O(pd )

Division and Slope Factorization of p-Adic Polynomials Introduction

Precision formulae

Proposition (addition)

k0 k1 min(k0,k1) (x0 + O(p )) + (x1 + O(p )) = x0 + x1 + O(p )

...... Proposition (division)

xpa + O(pb) = x ∗ y −1pa−c + O(pmin(d+a−2c,b−c)) ypc + O(pd ) In particular, 1 = y −1p−c + O(pd−2c ) pc y + O(pd )

Division and Slope Factorization of p-Adic Polynomials Introduction

Precision formulae

Proposition (addition)

k0 k1 min(k0,k1) (x0 + O(p )) + (x1 + O(p )) = x0 + x1 + O(p )

Proposition (multiplication)

k0 k1 min(k0+vp (x1),k1+vp (x0)) (x0 + O(p )) ∗ (x1 + O(p )) = x0 ∗ x1 + O(p )

...... Division and Slope Factorization of p-Adic Polynomials Introduction

Precision formulae

Proposition (addition)

k0 k1 min(k0,k1) (x0 + O(p )) + (x1 + O(p )) = x0 + x1 + O(p )

Proposition (multiplication)

k0 k1 min(k0+vp (x1),k1+vp (x0)) (x0 + O(p )) ∗ (x1 + O(p )) = x0 ∗ x1 + O(p )

Proposition (division)

xpa + O(pb) = x ∗ y −1pa−c + O(pmin(d+a−2c,b−c)) ypc + O(pd ) In particular, 1 = y −1p−c + O(pd−2c ) pc y + O(pd )

...... Modulus P = X 2 + 2. A = 5 − 2X + 4X 2. n Compute iteratively (naively) An = A mod P.

Behaviour of signiﬁcant digits n 100 101 102 103 104 105 106 relative precision of lc(An) 10 9 8 7 6 5 4

Question: How can we obtain satisfying / optimal behaviour regarding to precision?

Division and Slope Factorization of p-Adic Polynomials Introduction

Interesting behaviour

Large modular multiplication

Setting: 10 signiﬁcant digits in Q2.

...... n Compute iteratively (naively) An = A mod P.

Behaviour of signiﬁcant digits n 100 101 102 103 104 105 106 relative precision of lc(An) 10 9 8 7 6 5 4

Question: How can we obtain satisfying / optimal behaviour regarding to precision?

Division and Slope Factorization of p-Adic Polynomials Introduction

Interesting behaviour

Large modular multiplication

Setting: 10 signiﬁcant digits in Q2. Modulus P = X 2 + 2. A = 5 − 2X + 4X 2.

...... Behaviour of signiﬁcant digits n 100 101 102 103 104 105 106 relative precision of lc(An) 10 9 8 7 6 5 4

Question: How can we obtain satisfying / optimal behaviour regarding to precision?

Division and Slope Factorization of p-Adic Polynomials Introduction

Interesting behaviour

Large modular multiplication

Setting: 10 signiﬁcant digits in Q2. Modulus P = X 2 + 2. A = 5 − 2X + 4X 2. n Compute iteratively (naively) An = A mod P.

...... n 100 101 102 103 104 105 106 relative precision of lc(An) 10 9 8 7 6 5 4

Question: How can we obtain satisfying / optimal behaviour regarding to precision?

Division and Slope Factorization of p-Adic Polynomials Introduction

Interesting behaviour

Large modular multiplication

Setting: 10 signiﬁcant digits in Q2. Modulus P = X 2 + 2. A = 5 − 2X + 4X 2. n Compute iteratively (naively) An = A mod P.

Behaviour of signiﬁcant digits

...... Question: How can we obtain satisfying / optimal behaviour regarding to precision?

Division and Slope Factorization of p-Adic Polynomials Introduction

Interesting behaviour

Large modular multiplication

Setting: 10 signiﬁcant digits in Q2. Modulus P = X 2 + 2. A = 5 − 2X + 4X 2. n Compute iteratively (naively) An = A mod P.

Behaviour of signiﬁcant digits n 100 101 102 103 104 105 106 relative precision of lc(An) 10 9 8 7 6 5 4

...... Division and Slope Factorization of p-Adic Polynomials Introduction

Interesting behaviour

Large modular multiplication

Setting: 10 signiﬁcant digits in Q2. Modulus P = X 2 + 2. A = 5 − 2X + 4X 2. n Compute iteratively (naively) An = A mod P.

Behaviour of signiﬁcant digits n 100 101 102 103 104 105 106 relative precision of lc(An) 10 9 8 7 6 5 4

Question: How can we obtain satisfying / optimal behaviour regarding to precision?

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Table of contents

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... ∈ Qn ′ Let x 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.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision The Main lemma of p-adic diﬀerential precision

Lemma (CRV14) Qn → Qm Let f : p p be a (strictly) diﬀerentiable mapping.

...... Then for any ball B = B(0, r) small enough,

f (x + B) = f (x) + f ′(x) · B.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision The Main lemma of p-adic diﬀerential precision

Lemma (CRV14) Qn → Qm Let f : p p be a (strictly) diﬀerentiable mapping. ∈ Qn ′ Let x p. We assume that f (x) is surjective.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision The Main lemma of p-adic diﬀerential precision

Lemma (CRV14) Qn → Qm Let f : p p be a (strictly) diﬀerentiable mapping. ∈ Qn ′ Let x 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.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision The Main lemma of p-adic diﬀerential precision

Lemma (CRV14) Qn → Qm Let f : p p be a (strictly) diﬀerentiable mapping. ∈ Qn ′ Let x 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.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Geometrical meaning

Interpretation

x f (x)

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Geometrical meaning

Interpretation

x f (x)

f ′(x)

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Geometrical meaning

Interpretation

x f (x)

f ′(x)

′ B f (x) · B

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Geometrical meaning

Interpretation

x f (x) x + B

f ′(x)

′ B f (x) · B

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Geometrical meaning

Interpretation

x f (x) x + B

f f ′(x)

′ B f (x) · B

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Geometrical meaning

Interpretation

x f (x) x + B ′ f (x) + f (x) · B

f f ′(x)

′ B f (x) · B

...... Lemma Qn → Qm Let f : p p be a (strictly) diﬀerentiable mapping. ∈ Qn ′ Let x p. We assume that f (x) is surjective. Then for any ball B = B(0, r) small enough,

f (x + ) = f (x) + f ′(x) · .

Remark This allows more models of precision, like (x, y) = (1 + O(p10), 1 + O(p)).

Remark Our framework can be extended to (complete) ultrametric K-vector n m s spaces (e.g. being Fp((X)) , Q((X)) , R((ε)) ).

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Lattices

...... Remark This allows more models of precision, like (x, y) = (1 + O(p10), 1 + O(p)).

Remark Our framework can be extended to (complete) ultrametric K-vector n m s spaces (e.g. being Fp((X)) , Q((X)) , R((ε)) ).

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Lattices

Lemma Qn → Qm Let f : p p be a (strictly) diﬀerentiable mapping. ∈ Qn ′ Let x 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.

...... Remark This allows more models of precision, like (x, y) = (1 + O(p10), 1 + O(p)).

Remark Our framework can be extended to (complete) ultrametric K-vector n m s spaces (e.g. being Fp((X)) , Q((X)) , R((ε)) ).

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Lattices

Lemma Qn → Qm Let f : p p be a (strictly) diﬀerentiable mapping. ∈ Qn ′ Let x p. We assume that f (x) is surjective. Then for any ball B = B(0, r) small enough, for any open Zp-lattice H ⊂ B f (x + H) = f (x) + f ′(x) · H.

...... Remark Our framework can be extended to (complete) ultrametric K-vector n m s spaces (e.g. being Fp((X)) , Q((X)) , R((ε)) ).

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Lattices

Remark This allows more models of precision, like (x, y) = (1 + O(p10), 1 + O(p)).

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Lattices

Remark This allows more models of precision, like (x, y) = (1 + O(p10), 1 + O(p)).

Remark Our framework can be extended to (complete) ultrametric K-vector F n Q m R s spaces (e.g. being p((X)) , ((X)) , ((ε)) )...... What is small enough ? How can we determine when the lemma applies ? When f is locally analytic, it essentially corresponds to

∞ ∑+ 1 f (k)(x) · Hk ⊂ f ′(x) · H. k! k=2 This can be determined with Newton-polygon techniques.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Higher diﬀerentials

...... When f is locally analytic, it essentially corresponds to

∞ ∑+ 1 f (k)(x) · Hk ⊂ f ′(x) · H. k! k=2 This can be determined with Newton-polygon techniques.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Higher diﬀerentials

What is small enough ? How can we determine when the lemma applies ?

...... This can be determined with Newton-polygon techniques.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Higher diﬀerentials

What is small enough ? How can we determine when the lemma applies ? When f is locally analytic, it essentially corresponds to

∞ ∑+ 1 f (k)(x) · Hk ⊂ f ′(x) · H. k! k=2

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision p-Adic Precision Higher diﬀerentials

What is small enough ? How can we determine when the lemma applies ? When f is locally analytic, it essentially corresponds to

∞ ∑+ 1 f (k)(x) · Hk ⊂ f ′(x) · H. k! k=2 This can be determined with Newton-polygon techniques.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Study of the division Table of contents

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... Lattice and division

A and B are known with precision lattice HA and HB. Then (HQ, HR ) are given by the Euclidean division of HA − QHB by B.

Proof.

A = BQ + R, A + δA = (B + δB)(Q + δQ) + (R + δR), δA = QδB + BδQ + δR (at ﬁrst order), δA − QδB = BδQ + δR.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Study of the division Handling the precision

Lattice precision

To each polynomial, we attach a Zp-lattice (given by a basis of this lattice).

...... Proof.

A = BQ + R, A + δA = (B + δB)(Q + δQ) + (R + δR), δA = QδB + BδQ + δR (at ﬁrst order), δA − QδB = BδQ + δR.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Study of the division Handling the precision

Lattice precision

To each polynomial, we attach a Zp-lattice (given by a basis of this lattice).

Lattice and division

A and B are known with precision lattice HA and HB. Then (HQ, HR ) are given by the Euclidean division of HA − QHB by B.

...... δA = QδB + BδQ + δR (at ﬁrst order), δA − QδB = BδQ + δR.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Study of the division Handling the precision

Lattice precision

To each polynomial, we attach a Zp-lattice (given by a basis of this lattice).

Lattice and division

A and B are known with precision lattice HA and HB. Then (HQ, HR ) are given by the Euclidean division of HA − QHB by B.

Proof.

A = BQ + R, A + δA = (B + δB)(Q + δQ) + (R + δR),

...... δA − QδB = BδQ + δR.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Study of the division Handling the precision

Lattice precision

To each polynomial, we attach a Zp-lattice (given by a basis of this lattice).

Lattice and division

A and B are known with precision lattice HA and HB. Then (HQ, HR ) are given by the Euclidean division of HA − QHB by B.

Proof.

A = BQ + R, A + δA = (B + δB)(Q + δQ) + (R + δR), δA = QδB + BδQ + δR (at ﬁrst order),

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Study of the division Handling the precision

Lattice precision

To each polynomial, we attach a Zp-lattice (given by a basis of this lattice).

Lattice and division

A and B are known with precision lattice HA and HB. Then (HQ, HR ) are given by the Euclidean division of HA − QHB by B.

Proof.

A = BQ + R, A + δA = (B + δB)(Q + δQ) + (R + δR), δA = QδB + BδQ + δR (at ﬁrst order), δA − QδB = BδQ + δR.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Table of contents

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... For A × B, δ(A × B) = AδB + BδA.

For remainder R in A = BQ + R,

δ(R) = δA − QδB mod B.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Lattice precision and modular multiplication

Composed derivatives It is easy to handle in an optimal way the modular multiplication by applying:

...... δ(A × B) = AδB + BδA.

For remainder R in A = BQ + R,

δ(R) = δA − QδB mod B.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Lattice precision and modular multiplication

Composed derivatives It is easy to handle in an optimal way the modular multiplication by applying: For A × B,

...... For remainder R in A = BQ + R,

δ(R) = δA − QδB mod B.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Lattice precision and modular multiplication

Composed derivatives It is easy to handle in an optimal way the modular multiplication by applying: For A × B, δ(A × B) = AδB + BδA.

...... δ(R) = δA − QδB mod B.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Lattice precision and modular multiplication

Composed derivatives It is easy to handle in an optimal way the modular multiplication by applying: For A × B, δ(A × B) = AδB + BδA.

For remainder R in A = BQ + R,

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Lattice precision and modular multiplication

Composed derivatives It is easy to handle in an optimal way the modular multiplication by applying: For A × B, δ(A × B) = AδB + BδA.

For remainder R in A = BQ + R,

δ(R) = δA − QδB mod B.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Toward numerical understanding

Displaying precision

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Toward numerical understanding

Displaying precision ⊂ Qn Let H p be a lattice.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Toward numerical understanding

Displaying precision ⊂ Qn Let H p be a lattice.

H0 = π1(H) ⊕ · · · ⊕ πn(H) provides the best precision on coordinates. It is the smallest diagonal lattice containing H.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Toward numerical understanding

Displaying precision ⊂ Qn Let H p be a lattice.

H0 = π1(H) ⊕ · · · ⊕ πn(H) provides the best precision on coordinates. It is the smallest diagonal lattice containing H.

Diﬀused digits The number of diﬀused digits of precision of H is the length of H0/H.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication More on diﬀused digits

Diﬀused digits The number of diﬀused digits of precision of H is the length of H0/H.

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication More on diﬀused digits

Diﬀused digits The number of diﬀused digits of precision of H is the length of H0/H.

Another deﬁnition Here, the number of diﬀused digits is:

− | | logp ( H0/H ) .

...... Then   1   H0 = p . p2

Number of diﬀused digits   1   The SNF of H0/H is: p . Hence 2 diﬀused digits. p

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Diﬀused digits: example

Diﬀused digits: the lattice Let ⟨     ⟩ 1 p 2p H =  p  ,  p2  ,  p2  . p2 2p4 2p3

...... Number of diﬀused digits   1   The SNF of H0/H is: p . Hence 2 diﬀused digits. p

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Diﬀused digits: example

Diﬀused digits: the lattice Let ⟨     ⟩ 1 p 2p H =  p  ,  p2  ,  p2  . p2 2p4 2p3 Then   1   H0 = p . p2

...... Hence 2 diﬀused digits.

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Diﬀused digits: example

Diﬀused digits: the lattice Let ⟨     ⟩ 1 p 2p H =  p  ,  p2  ,  p2  . p2 2p4 2p3 Then   1   H0 = p . p2

Number of diﬀused digits   1   The SNF of H0/H is: p . p

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Diﬀused digits: example

Diﬀused digits: the lattice Let ⟨     ⟩ 1 p 2p H =  p  ,  p2  ,  p2  . p2 2p4 2p3 Then   1   H0 = p . p2

Number of diﬀused digits   1   The SNF of H0/H is: p . Hence 2 diﬀused digits. p

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication ∏ n Comparison: i=1 Ai mod M

Gain of precision Modulus M n Jagged Lattice (not dif. + dif.) 10 0.2 0.2 + 0.0 X 5 + X 2 + 1 50 4.2 4.2 + 0.0 (Irred. mod 2) 100 11.2 11.2 + 0.0 10 0.4 0.9 + 6.0 X 5 + 1 50 5.6 11.1 + 42.0 (Sep. mod 2) 100 13.6 27.0 + 87.0 10 6.2 6.2 + 0.0 X 5 + 2 50 44.0 44.0 + 0.0 (Eisenstein) 100 92.5 92.5 + 0.0 10 0.6 4.7 + 1.4 (X + 1)5 + 2 50 7.1 42.6 + 1.4 (Shift Eisenstein) 100 15.1 91.8 + 1.4 10 1.7 7.9 + 9.8 X 5 + X + 2 50 8.1 70.7 + 59.8 (Two slopes) 100 16.1 152.6 + 125.9 Figure: Precision for modular multiplication

...... N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

...... Zp[X] Zp[X]

×Ai

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

. .no. . diﬀused...... digits...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: two possibilities

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

×Ai

. . . .diﬀused...... digits...... Zp[X] Zp[X]

without lattice with lattice

Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 0 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 1 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 2 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 3 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 3 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 4 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 5 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 5 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 10 ...... Division and Slope Factorization of p-Adic Polynomials Division and Diﬀerential Precision Modular Multiplication Qualitative understanding: long-term

N Input: P, A1,..., An ∈ Zp[X]d known at precision O(p ) Output: the product A1A2 ··· An mod P 1. A = 1 + O(pN ) 2. for i in 1, 2,... n:

3. A = A · Ai 4. return A

Zp[X] Zp[X]

without lattice with lattice i = 100 ...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Table of contents

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... Let U = {(0, v(a0)),..., (n, v(an))}. We deﬁne the Newton polygon of f , NP(f ), as the lower convex hull of U. By lower convex hull, we mean the points of the convex hull of U below the straight line from (0, (a0)) to (n, v(an)).

Proposition

Let Newtf be the (point-wise) biggest convex mapping below U. Then the graph of Newtf is (the lower frontier of) NP(f ).

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Newton polygon of a polynomial

Deﬁnition n Let f (X) = a0 + ··· + anX ∈ Qp[x] (remark: QpJxK would also be ﬁne).

...... We deﬁne the Newton polygon of f , NP(f ), as the lower convex hull of U. By lower convex hull, we mean the points of the convex hull of U below the straight line from (0, (a0)) to (n, v(an)).

Proposition

Let Newtf be the (point-wise) biggest convex mapping below U. Then the graph of Newtf is (the lower frontier of) NP(f ).

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Newton polygon of a polynomial

Deﬁnition n Let f (X) = a0 + ··· + anX ∈ Qp[x] (remark: QpJxK would also be ﬁne). Let U = {(0, v(a0)),..., (n, v(an))}.

...... By lower convex hull, we mean the points of the convex hull of U below the straight line from (0, (a0)) to (n, v(an)).

Proposition

Let Newtf be the (point-wise) biggest convex mapping below U. Then the graph of Newtf is (the lower frontier of) NP(f ).

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Newton polygon of a polynomial

...... Proposition

Let Newtf be the (point-wise) biggest convex mapping below U. Then the graph of Newtf is (the lower frontier of) NP(f ).

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Newton polygon of a polynomial

Definition n Let f (X) = a0 + ··· + anX ∈ Qp[x] (remark: QpJxK would also be fine). Let U = {(0, v(a0)),..., (n, v(an))}. We define the Newton polygon of f , NP(f ), as the lower convex hull of U. By lower convex hull, we mean the points of the convex hull of U below the straight line from (0, (a0)) to (n, v(an)).

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Newton polygon of a polynomial

Proposition

Let Newtf be the (point-wise) biggest convex mapping below U. Then the graph of Newtf is (the lower frontier of) NP(f ).

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3

0 1 2 3 4

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3

1 NP(P)

0 1 2 3 4

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Vocabulary

Deﬁnition ′ A slope of the Newton polygon of f is an element of Newtf ([0, n]).

Deﬁnition

If λ is a slope of Newtf , we call segment of slope λ of Newtf the set { ⧸ ′ } (x, Newtf (x)) Newtf (x) = λ .

Deﬁnition The length of this slope is the length of its projection on the x-axis.

...... Moreover, the number of roots of f (with multiplicity) of valuation λ, is the length of the segment of slope −λ of Newtf .

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Fundamental theorem of Newton polygons

Theorem

f has a root of valuation λ iﬀ −λ is a slope of Newtf .

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Fundamental theorem of Newton polygons

Theorem

f has a root of valuation λ iﬀ −λ is a slope of Newtf . Moreover, the number of roots of f (with multiplicity) of valuation λ, is the length of the segment of slope −λ of Newtf .

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3

1 NP(P)

0 1 2 3 4

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3

One slope for 0 of length 2, one slope 1/2 of length 2. 2

1 NP(P)

0 1 2 3 4

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3

One slope for 0 of length 2, one slope 1/2 of length 2. 2 Two roots of valuation 0, two of valuation −1/2. 1 NP(P)

0 1 2 3 4

...... Proposition (Multiplicativity) If f and g are two polynomials, then the Newton polygon of fg has for slopes that of f and g, with length the sum of that of f and g.

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Basic operations

Proposition (Addition) If f and g are two polynomials, then the Newton polygon of f + g can be lower-bounded by taking the lower convex hull for the vertices of Newtf and Newtg .

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Basic operations

Proposition (Addition) If f and g are two polynomials, then the Newton polygon of f + g can be lower-bounded by taking the lower convex hull for the vertices of Newtf and Newtg .

Proposition (Multiplicativity) If f and g are two polynomials, then the Newton polygon of fg has for slopes that of f and g, with length the sum of that of f and g.

...... Corollary

If NewtP has more than one slope, P is not irreducible.

Remark There are good irreducibility criterion based on testing whether one slope can be obtained by multiplication (namely, Dumas and Eisenstein).

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Some remarks

Proposition

If P ∈ Qp[X] is irreducible, then all its roots have the same valuation. Hence, NewtP has only one slope.

Remark

The converse is false. For instance, (X − 1)(X − 2) over Q5.

...... Remark There are good irreducibility criterion based on testing whether one slope can be obtained by multiplication (namely, Dumas and Eisenstein).

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Some remarks

Proposition

If P ∈ Qp[X] is irreducible, then all its roots have the same valuation. Hence, NewtP has only one slope.

Remark

The converse is false. For instance, (X − 1)(X − 2) over Q5.

Corollary

If NewtP has more than one slope, P is not irreducible.

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Basics Some remarks

Proposition

If P ∈ Qp[X] is irreducible, then all its roots have the same valuation. Hence, NewtP has only one slope.

Remark

The converse is false. For instance, (X − 1)(X − 2) over Q5.

Corollary

If NewtP has more than one slope, P is not irreducible.

Remark There are good irreducibility criterion based on testing whether one slope can be obtained by multiplication (namely, Dumas and Eisenstein).

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Table of contents

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

What is the Newton polygon of the remainder in the division of A by B (in Qp[X])? What is the Newton polygon of the quotient in the division of A by B (in Qp[X])?

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation

NP(B) puX n u

d − 1 d n order

Division of puX n by B : puX n = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation

NP(B) puX n u

d − 1 d n order

Division of puX n by B : puX n = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation

NP(B) puX n u

d − 1 d n order

Division of puX n by B : puX n = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation

NP(B) puX n u

d − 1 d n order

Division of puX n by B : puX n = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation NP(R)

NP(B) puX n u

d − 1 d n order

Division of puX n by B : puX n = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation NP(R)

NP(B) puX n u

d − 1 d n order

Division of puX n by B : puX n = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation NP(R) NP(Q)

NP(B) puX n u

d − 1 d n − d n order

Division of puX n by B : puX n = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation

NP(A) NP(B)

d − 1 d order

Division of A by B : A = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation

NP(A) NP(B)

d − 1 d order

Division of A by B : A = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Euclidean division Euclidean division and Newton polygon

Lemma (Division lemma)

valuation

NP(R) NP(Q) NP(A) NP(B)

d − 1 d order

Division of A by B : A = BQ + R

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Precision: Return on Modular Multiplication Table of contents

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... Newton precision For A = BQ + R with A known with precision-polygon φ, we can apply the previous construction to φ divided by B to obtain the precision on Q and R.

Remark This proved to be useful to handle precision for the computation of the characteristic polynomial.

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Precision: Return on Modular Multiplication Handling the precision

Lattice precision

A and B are known with precision lattice HA and HB. Then (HQ, HR ) are given by the Euclidean division of HA − QHB by B.

...... Remark This proved to be useful to handle precision for the computation of the characteristic polynomial.

Division and Slope Factorization of p-Adic Polynomials Newton Polygons Precision: Return on Modular Multiplication Handling the precision

Lattice precision

A and B are known with precision lattice HA and HB. Then (HQ, HR ) are given by the Euclidean division of HA − QHB by B.

Newton precision For A = BQ + R with A known with precision-polygon φ, we can apply the previous construction to φ divided by B to obtain the precision on Q and R.

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Precision: Return on Modular Multiplication Handling the precision

Lattice precision

A and B are known with precision lattice HA and HB. Then (HQ, HR ) are given by the Euclidean division of HA − QHB by B.

Newton precision For A = BQ + R with A known with precision-polygon φ, we can apply the previous construction to φ divided by B to obtain the precision on Q and R.

Remark This proved to be useful to handle precision for the computation of the characteristic polynomial.

...... Division and Slope Factorization of p-Adic Polynomials Newton Polygons Precision: Return on Modular Multiplication ∏ n Comparison: i=1 Ai mod M

Gain of precision Modulus M n Jagged Newton Lattice (not dif. + dif.) 10 0.2 0.2 0.2 + 0.0 X 5 + X 2 + 1 50 4.2 4.2 4.2 + 0.0 (Irred. mod 2) 100 11.2 11.2 11.2 + 0.0 10 0.4 0.4 0.9 + 6.0 X 5 + 1 50 5.6 5.6 11.1 + 42.0 (Sep. mod 2) 100 13.6 13.6 27.0 + 87.0 10 6.2 6.2 6.2 + 0.0 X 5 + 2 50 44.0 44.0 44.0 + 0.0 (Eisenstein) 100 92.5 92.5 92.5 + 0.0 10 0.6 0.6 4.7 + 1.4 (X + 1)5 + 2 50 7.1 7.1 42.6 + 1.4 (Shift Eisenstein) 100 15.1 15.1 91.8 + 1.4 10 1.7 1.7 7.9 + 9.8 X 5 + X + 2 50 8.1 8.1 70.7 + 59.8 (Two slopes) 100 16.1 16.1 152.6 + 125.9 Figure: Precision for modular multiplication

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Table of contents

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... ∏ We can write f = i fi . The fi ’s are all of one slope. They have respectively diﬀerent slopes.

Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Factoring respecting slopes

Theorem

Let f ∈ Qp[X]. Then,

...... The fi ’s are all of one slope. They have respectively diﬀerent slopes.

Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Factoring respecting slopes

Theorem Let f ∈ Q [X]. Then, p ∏ We can write f = i fi .

...... They have respectively diﬀerent slopes.

Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Factoring respecting slopes

Theorem Let f ∈ Q [X]. Then, p ∏ We can write f = i fi . The fi ’s are all of one slope.

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Factoring respecting slopes

Theorem Let f ∈ Q [X]. Then, p ∏ We can write f = i fi . The fi ’s are all of one slope. They have respectively diﬀerent slopes.

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3.

2 NP(P) 1

0 1 2 3 4

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3.

2 NP(P)

1 NP(f2)

NP(f1)

0 1 2 3 4

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3.

P = (2 + 3X + X 2) × (1 + 3X 2) 2 NP(P)

1 NP(f2)

NP(f1)

0 1 2 3 4

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme 例

A Newton polygon

2 3 4 P = 2 + 3X + 7X + 9X + 3X , over Q3.

P = (2 + 3X + X 2) × (1 + 3X 2) 2 NP(P) Remark: 2 + 3X + X 2 = (1 + X)(1 + 2X).

1 NP(f2)

NP(f1)

0 1 2 3 4

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme A Newton iteration

The iteration Already found in Polynomial root ﬁnding over local rings and application to error correcting codes by Berthomieu, Lecerf, Quintin:

Ai+1 := Ai + (Vi P mod Ai )

Bi+1 := P \quo Ai+1 − 2 Vi+1 := (2Vi Vi Bi+1) mod Ai+1

The result

Ai , Bi , Vi converge quadratically to A, B, V such that AB = P, V is the inverse of B modulo A and A, B have the desired Newton polygons.

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Ideas on the proof

About the proof

We monitor Ri = Ai+1 − Ai and Si = P mod Ai .

We prove that both NP(Ri ) and NP(Si ) goes to inﬁnity.

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step.

NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step. Euclidean division.

NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step. Euclidean division.

NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step. Euclidean division.

NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step. Euclidean division.

NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step. Euclidean division.

NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step.

NP(R0) NP(S0) NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step.

NP(R0) NP(S0) NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

First step.

NP(R0) NP(S0) NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

Second Step.

NF (P)

λ 1 = slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

Second Step.

NP(R1) NP(S1) NF (P)

λ 1 = 2κ slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

Second Step.

NP(R1) NP(S1) NF (P)

λ 1 = 2κ slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

Third Step.

NP(R2) NP(S2) NF (P)

4κ λ 1 = 2κ slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization A Newton scheme Illustration

Third Step.

NP(R2) NP(S2) NF (P)

4κ λ 1 = 2κ slope φ κ = λ0 A0 slope

0 d−1 d d+1

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision Table of contents

1 Division and Diﬀerential Precision p-Adic Precision Study of the division Modular Multiplication

2 Newton Polygons Basics Euclidean division Precision: Return on Modular Multiplication

3 Slope factorization A Newton scheme Applying diﬀerential precision

...... Diﬀerential (1) 1 The application FA : P 7→ A is of class C . Its diﬀerential at some point P is the linear mapping

dP 7→ dA(1) = (V (1) dP) mod A(1)

where A(1)B(1) = P and V (1) is the inverse of B(1) modulo A(1).

Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision What about precision?

Setting (1) (1) (1) Let FA : P 7→ A be the application such that P = A B , with A(1), B(1) corresponding to the slopes before/after the breakpoint d.

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision What about precision?

Setting (1) (1) (1) Let FA : P 7→ A be the application such that P = A B , with A(1), B(1) corresponding to the slopes before/after the breakpoint d.

Diﬀerential (1) 1 The application FA : P 7→ A is of class C . Its diﬀerential at some point P is the linear mapping

dP 7→ dA(1) = (V (1) dP) mod A(1)

where A(1)B(1) = P and V (1) is the inverse of B(1) modulo A(1).

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision Some numerical results: A 7→ AB 7→ A.

Mean gain of precision Polynomials Precision Jagged Newton Lattice A B absolute −14.5 −14.7 0.0 d relative −1.5 −3.6 0.0 A B absolute 0.0 0.0 0.0 d relative −0.3 −1.2 0.0

...... Step-by-step analysis: as a first step. Can show differentiability and naïve loss in precision during the computation. Differential calculus: intrinsic and can handle both gain and loss. Lattice precision: achieving and understandig the best precision.

On polynomial computations Diﬀused digits for modular multiplication. Newton iteration for slope factorisation.

Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision To sum up

On p-adic precision

...... Diﬀerential calculus: intrinsic and can handle both gain and loss. Lattice precision: achieving and understandig the best precision.

On polynomial computations Diﬀused digits for modular multiplication. Newton iteration for slope factorisation.

Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision To sum up

On p-adic precision Step-by-step analysis: as a ﬁrst step. Can show diﬀerentiability and naïve loss in precision during the computation.

...... Lattice precision: achieving and understandig the best precision.

On polynomial computations Diﬀused digits for modular multiplication. Newton iteration for slope factorisation.

Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision To sum up

On p-adic precision Step-by-step analysis: as a first step. Can show differentiability and naïve loss in precision during the computation. Differential calculus: intrinsic and can handle both gain and loss.

...... On polynomial computations Diﬀused digits for modular multiplication. Newton iteration for slope factorisation.

Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision To sum up

...... Diﬀused digits for modular multiplication. Newton iteration for slope factorisation.

Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision To sum up

On polynomial computations

...... Newton iteration for slope factorisation.

Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision To sum up

On polynomial computations Diﬀused digits for modular multiplication.

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision To sum up

On polynomial computations Diﬀused digits for modular multiplication. Newton iteration for slope factorisation.

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision References

Initial article Xavier Caruso, David Roe and Tristan Vaccon Tracking p-adic precision, ANTS XI, 2014.

Linear Algebra Xavier Caruso, David Roe and Tristan Vaccon p-adic stability in linear algebra, ISSAC 2015.

Polynomial Computations Xavier Caruso, David Roe and Tristan Vaccon Division and Slope Factorization of p-Adic Polynomials, ISSAC 2016.

...... Division and Slope Factorization of p-Adic Polynomials Slope factorization Applying diﬀerential precision Thank you for your attention

Thanks ′ ′ x + O(pN ) y + O(pM ) ⊂ f (x) + O(pN )

x x + B ′ f (x) + f (x) · B

f f ′(x)

′ B f (x) · B

......