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Division and Slope Factorization of p-Adic Polynomials
Division and Slope Factorization of p-AdicPolynomials
Xavier Caruso, David Roe Tristan Vaccon
Univ.Rennes 1, Univ. Pittsburgh, 立教大学
July 22nd, 2016
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Why should one work with p-adic numbers ?
p-adic methodsWorking in Qp instead of Q, one can handle more efficiently thecoefficients growth ;
e.g. Dixon’s method (used in F4), Polynomial factorization viaHensel’s lemma.
p-adic algorithmsGoing from Z/pZ to Zp and then back to Z/pZ enables morecomputation, 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-adiccohomology ;
My personal (long-term) motivationComputing (some) moduli spaces of p-adic Galois representations.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Why should one work with p-adic numbers ?
p-adic methodsWorking in Qp instead of Q, one can handle more efficiently thecoefficients growth ;e.g. Dixon’s method (used in F4), Polynomial factorization viaHensel’s lemma.
p-adic algorithmsGoing from Z/pZ to Zp and then back to Z/pZ enables morecomputation, 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-adiccohomology ;
My personal (long-term) motivationComputing (some) moduli spaces of p-adic Galois representations.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Why should one work with p-adic numbers ?
p-adic methodsWorking in Qp instead of Q, one can handle more efficiently thecoefficients growth ;e.g. Dixon’s method (used in F4), Polynomial factorization viaHensel’s lemma.
p-adic algorithmsGoing from Z/pZ to Zp and then back to Z/pZ enables morecomputation,
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-adiccohomology ;
My personal (long-term) motivationComputing (some) moduli spaces of p-adic Galois representations.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Why should one work with p-adic numbers ?
p-adic methodsWorking in Qp instead of Q, one can handle more efficiently thecoefficients growth ;e.g. Dixon’s method (used in F4), Polynomial factorization viaHensel’s lemma.
p-adic algorithmsGoing from Z/pZ to Zp and then back to Z/pZ enables morecomputation, 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-adiccohomology ;
My personal (long-term) motivationComputing (some) moduli spaces of p-adic Galois representations.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Why should one work with p-adic numbers ?
p-adic methodsWorking in Qp instead of Q, one can handle more efficiently thecoefficients growth ;e.g. Dixon’s method (used in F4), Polynomial factorization viaHensel’s lemma.
p-adic algorithmsGoing from Z/pZ to Zp and then back to Z/pZ enables morecomputation, 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-adiccohomology ;
My personal (long-term) motivationComputing (some) moduli spaces of p-adic Galois representations.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Why should one work with p-adic numbers ?
p-adic methodsWorking in Qp instead of Q, one can handle more efficiently thecoefficients growth ;e.g. Dixon’s method (used in F4), Polynomial factorization viaHensel’s lemma.
p-adic algorithmsGoing from Z/pZ to Zp and then back to Z/pZ enables morecomputation, 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-adiccohomology ;
My personal (long-term) motivationComputing (some) moduli spaces of p-adic Galois representations.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Studying polynomial computations over p-adics
A building blockAt ISSAC 2015, we have studied the p-adic stability of somecomputations in linear algebra.
This year, we study basic operationsrelated to polynomial computations.
More motivationsUnderstanding basic operations related to field extensions, inparticular division and quotients.Understanding the behaviour of precision during factorisation: overQp or kJT K, or as an intermediate to factorisation over Q.
Today’s highlightsOptimal tracking of precision for modular multiplication anddiffused digits.Precision for slope factorization algorithms.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Studying polynomial computations over p-adics
A building blockAt ISSAC 2015, we have studied the p-adic stability of somecomputations in linear algebra. This year, we study basic operationsrelated to polynomial computations.
More motivationsUnderstanding basic operations related to field extensions, inparticular division and quotients.Understanding the behaviour of precision during factorisation: overQp or kJT K, or as an intermediate to factorisation over Q.
Today’s highlightsOptimal tracking of precision for modular multiplication anddiffused digits.Precision for slope factorization algorithms.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Studying polynomial computations over p-adics
A building blockAt ISSAC 2015, we have studied the p-adic stability of somecomputations in linear algebra. This year, we study basic operationsrelated to polynomial computations.
More motivationsUnderstanding basic operations related to field extensions, inparticular division and quotients.
Understanding the behaviour of precision during factorisation: overQp or kJT K, or as an intermediate to factorisation over Q.
Today’s highlightsOptimal tracking of precision for modular multiplication anddiffused digits.Precision for slope factorization algorithms.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Studying polynomial computations over p-adics
A building blockAt ISSAC 2015, we have studied the p-adic stability of somecomputations in linear algebra. This year, we study basic operationsrelated to polynomial computations.
More motivationsUnderstanding basic operations related to field extensions, inparticular division and quotients.Understanding the behaviour of precision during factorisation: overQp or kJT K, or as an intermediate to factorisation over Q.
Today’s highlightsOptimal tracking of precision for modular multiplication anddiffused digits.Precision for slope factorization algorithms.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Studying polynomial computations over p-adics
A building blockAt ISSAC 2015, we have studied the p-adic stability of somecomputations in linear algebra. This year, we study basic operationsrelated to polynomial computations.
More motivationsUnderstanding basic operations related to field extensions, inparticular division and quotients.Understanding the behaviour of precision during factorisation: overQp or kJT K, or as an intermediate to factorisation over Q.
Today’s highlightsOptimal tracking of precision for modular multiplication anddiffused digits.
Precision for slope factorization algorithms.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Studying polynomial computations over p-adics
A building blockAt ISSAC 2015, we have studied the p-adic stability of somecomputations in linear algebra. This year, we study basic operationsrelated to polynomial computations.
More motivationsUnderstanding basic operations related to field extensions, inparticular division and quotients.Understanding the behaviour of precision during factorisation: overQp or kJT K, or as an intermediate to factorisation over Q.
Today’s highlightsOptimal tracking of precision for modular multiplication anddiffused digits.Precision for slope factorization algorithms.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
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 applyingSchoolBook 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 ap-adic integer.
The p-adic integers form a subring Zp of Qp.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
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 applyingSchoolBook 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 ap-adic integer.
The p-adic integers form a subring Zp of Qp.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
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 applyingSchoolBook 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 ap-adic integer.
The p-adic integers form a subring Zp of Qp.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
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 applyingSchoolBook 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 ap-adic integer.
The p-adic integers form a subring Zp of Qp.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
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 applyingSchoolBook 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 ap-adic integer.
The p-adic integers form a subring Zp of Qp.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
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 applyingSchoolBook 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 ap-adic integer.
The p-adic integers form a subring Zp of Qp.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
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 applyingSchoolBook 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 ap-adic integer.
The p-adic integers form a subring Zp of Qp.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
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 applyingSchoolBook 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 ap-adic integer.
The p-adic integers form a subring Zp of Qp.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Summary on p-adics
Proposition
Zp/pZp = Z/pZ.
∀k ∈ N,Zp/pkZp = Z/pkZ.
A first ideaQp is an extension of Q where one can perform calculus, as simplyas over R.We are closer to arithmetic : we can reduce modulo p.
Remark
Qp R Zp Zp
Q Z/pZ Z/pZ
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Summary on p-adics
Proposition
Zp/pZp = Z/pZ.
∀k ∈ N,Zp/pkZp = Z/pkZ.
A first ideaQp is an extension of Q where one can perform calculus, as simplyas over R.We are closer to arithmetic : we can reduce modulo p.
Remark
Qp R Zp Zp
Q Z/pZ Z/pZ
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Summary on p-adics
Proposition
Zp/pZp = Z/pZ.
∀k ∈ N,Zp/pkZp = Z/pkZ.
A first ideaQp is an extension of Q where one can perform calculus, as simplyas over R.
We are closer to arithmetic : we can reduce modulo p.
Remark
Qp R Zp Zp
Q Z/pZ Z/pZ
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Summary on p-adics
Proposition
Zp/pZp = Z/pZ.
∀k ∈ N,Zp/pkZp = Z/pkZ.
A first ideaQp is an extension of Q where one can perform calculus, as simplyas over R.We are closer to arithmetic : we can reduce modulo p.
Remark
Qp R Zp Zp
Q Z/pZ Z/pZ
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Summary on p-adics
Proposition
Zp/pZp = Z/pZ.
∀k ∈ N,Zp/pkZp = Z/pkZ.
A first ideaQp is an extension of Q where one can perform calculus, as simplyas over R.We are closer to arithmetic : we can reduce modulo p.
Remark
Qp R Zp Zp
Q Z/pZ Z/pZ
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Definition of the precision
Finite-precision p-adics
Elements of Qp can be written∑+∞
i=k aipi , with ai ∈ J0, p − 1K, k ∈ Zand p a prime number.While working with a computer, we usually only can consider thebeginning of this power serie expansion: we only consider elements of thefollowing form
∑d−1i=l aipi + O(pd) , with l ∈ Z.
Definition
The order, or the absolute precision of∑d−1
i=k aipi + O(pd) is d .
ExampleThe order of 3 ∗ 7−1 + 4 ∗ 70 + 5 ∗ 71 + 6 ∗ 72 + O(73) is 3.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Definition of the precision
Finite-precision p-adics
Elements of Qp can be written∑+∞
i=k aipi , with ai ∈ J0, p − 1K, k ∈ Zand p a prime number.While working with a computer, we usually only can consider thebeginning of this power serie expansion: we only consider elements of thefollowing form
∑d−1i=l aipi + O(pd) , with l ∈ Z.
Definition
The order, or the absolute precision of∑d−1
i=k aipi + O(pd) is d .
ExampleThe order of 3 ∗ 7−1 + 4 ∗ 70 + 5 ∗ 71 + 6 ∗ 72 + O(73) is 3.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Definition of the precision
Finite-precision p-adics
Elements of Qp can be written∑+∞
i=k aipi , with ai ∈ J0, p − 1K, k ∈ Zand p a prime number.While working with a computer, we usually only can consider thebeginning of this power serie expansion: we only consider elements of thefollowing form
∑d−1i=l aipi + O(pd) , with l ∈ Z.
Definition
The order, or the absolute precision of∑d−1
i=k aipi + O(pd) is d .
ExampleThe order of 3 ∗ 7−1 + 4 ∗ 70 + 5 ∗ 71 + 6 ∗ 72 + O(73) is 3.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Definition of the precision
Finite-precision p-adics
Elements of Qp can be written∑+∞
i=k aipi , with ai ∈ J0, p − 1K, k ∈ Zand p a prime number.While working with a computer, we usually only can consider thebeginning of this power serie expansion: we only consider elements of thefollowing form
∑d−1i=l aipi + O(pd) , with l ∈ Z.
Definition
The order, or the absolute precision of∑d−1
i=k aipi + O(pd) is d .
ExampleThe order of 3 ∗ 7−1 + 4 ∗ 70 + 5 ∗ 71 + 6 ∗ 72 + O(73) is 3.
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
p-adic precion vs real precisionThe 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 isa + b.
RemarkIt is quite the opposite to when dealing with real numbers, because ofRound-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
known up to 10(−n +1 ).
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
p-adic precion vs real precisionThe 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 isa + b.
RemarkIt is quite the opposite to when dealing with real numbers, because ofRound-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
known up to 10(−n +1 ).
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
p-adic precion vs real precisionThe 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 isa + b.
RemarkIt is quite the opposite to when dealing with real numbers, because ofRound-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
known up to 10(−n +1 ).
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
p-adic precion vs real precisionThe 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 isa + b.
RemarkIt is quite the opposite to when dealing with real numbers, because ofRound-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
known up to 10(−n +1 ).
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Precision formulae
Proposition (addition)
(x0 + O(pk0)) + (x1 + O(pk1)) = x0 + x1 + O(pmin(k0,k1))
Proposition (multiplication)
(x0 + O(pk0)) ∗ (x1 + O(pk1)) = x0 ∗ x1 + O(pmin(k0+vp(x1),k1+vp(x0)))
Proposition (division)
xpa + O(pb)ypc + O(pd) = x ∗ y−1pa−c + O(pmin(d+a−2c,b−c))
In particular, 1pcy + O(pd) = y−1p−c + O(pd−2c)
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Precision formulae
Proposition (addition)
(x0 + O(pk0)) + (x1 + O(pk1)) = x0 + x1 + O(pmin(k0,k1))
Proposition (multiplication)
(x0 + O(pk0)) ∗ (x1 + O(pk1)) = x0 ∗ x1 + O(pmin(k0+vp(x1),k1+vp(x0)))
Proposition (division)
xpa + O(pb)ypc + O(pd) = x ∗ y−1pa−c + O(pmin(d+a−2c,b−c))
In particular, 1pcy + O(pd) = y−1p−c + O(pd−2c)
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Precision formulae
Proposition (addition)
(x0 + O(pk0)) + (x1 + O(pk1)) = x0 + x1 + O(pmin(k0,k1))
Proposition (multiplication)
(x0 + O(pk0)) ∗ (x1 + O(pk1)) = x0 ∗ x1 + O(pmin(k0+vp(x1),k1+vp(x0)))
Proposition (division)
xpa + O(pb)ypc + O(pd) = x ∗ y−1pa−c + O(pmin(d+a−2c,b−c))
In particular, 1pcy + O(pd) = y−1p−c + O(pd−2c)
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Interesting behaviour
Large modular multiplicationSetting: 10 significant digits in Q2.
Modulus P = X 2 + 2.
A = 5 − 2X + 4X 2.
Compute iteratively (naively) An = An 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 toprecision?
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Interesting behaviour
Large modular multiplicationSetting: 10 significant digits in Q2.
Modulus P = X 2 + 2.
A = 5 − 2X + 4X 2.
Compute iteratively (naively) An = An 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 toprecision?
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Interesting behaviour
Large modular multiplicationSetting: 10 significant digits in Q2.
Modulus P = X 2 + 2.
A = 5 − 2X + 4X 2.
Compute iteratively (naively) An = An 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 toprecision?
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Interesting behaviour
Large modular multiplicationSetting: 10 significant digits in Q2.
Modulus P = X 2 + 2.
A = 5 − 2X + 4X 2.
Compute iteratively (naively) An = An 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 toprecision?
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Interesting behaviour
Large modular multiplicationSetting: 10 significant digits in Q2.
Modulus P = X 2 + 2.
A = 5 − 2X + 4X 2.
Compute iteratively (naively) An = An 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 toprecision?
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Division and Slope Factorization of p-Adic PolynomialsIntroduction
Interesting behaviour
Large modular multiplicationSetting: 10 significant digits in Q2.
Modulus P = X 2 + 2.
A = 5 − 2X + 4X 2.
Compute iteratively (naively) An = An 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 toprecision?
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Table of contents
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
The Main lemma of p-adic differential precision
Lemma (CRV14)Let f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
The Main lemma of p-adic differential precision
Lemma (CRV14)Let f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
The Main lemma of p-adic differential precision
Lemma (CRV14)Let f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
The Main lemma of p-adic differential precision
Lemma (CRV14)Let f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Geometrical meaning
Interpretation
x
B
f (x)
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Geometrical meaning
Interpretation
x
B
f (x)
f ′(x)
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Geometrical meaning
Interpretation
x
B
f (x)
f ′(x)
f ′(x) · B
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Geometrical meaning
Interpretation
xx + B
B
f (x)
f ′(x)
f ′(x) · B
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Geometrical meaning
Interpretation
xx + B
f
B
f (x)
f ′(x)
f ′(x) · B
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Geometrical meaning
Interpretation
xx + B
f
B
f (x)
f ′(x)
f (x) + f ′(x) · B
f ′(x) · B
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Lattices
LemmaLet f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. We assume that f ′(x) is surjective.
Then for any ball B = B(0, r) small enough,
f (x + ) = f (x) + f ′(x) · .
RemarkThis allows more models of precision, like
(x , y) = (1 + O(p10), 1 + O(p)).
RemarkOur framework can be extended to (complete) ultrametric K -vectorspaces (e.g. being Fp((X ))n, Q((X ))m, R((ε))s).
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Lattices
LemmaLet f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. 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.
RemarkThis allows more models of precision, like
(x , y) = (1 + O(p10), 1 + O(p)).
RemarkOur framework can be extended to (complete) ultrametric K -vectorspaces (e.g. being Fp((X ))n, Q((X ))m, R((ε))s).
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Lattices
LemmaLet f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. We assume that f ′(x) is surjective.
Then for any ball B = B(0, r) small enough, for any open Zp-latticeH ⊂ B
f (x + H) = f (x) + f ′(x) · H.
RemarkThis allows more models of precision, like
(x , y) = (1 + O(p10), 1 + O(p)).
RemarkOur framework can be extended to (complete) ultrametric K -vectorspaces (e.g. being Fp((X ))n, Q((X ))m, R((ε))s).
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Lattices
LemmaLet f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. We assume that f ′(x) is surjective.
Then for any ball B = B(0, r) small enough, for any open Zp-latticeH ⊂ B
f (x + H) = f (x) + f ′(x) · H.
RemarkThis allows more models of precision, like
(x , y) = (1 + O(p10), 1 + O(p)).
RemarkOur framework can be extended to (complete) ultrametric K -vectorspaces (e.g. being Fp((X ))n, Q((X ))m, R((ε))s).
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
p-Adic Precision
Lattices
LemmaLet f : Qn
p → Qmp be a (strictly) differentiable mapping.
Let x ∈ Qnp. We assume that f ′(x) is surjective.
Then for any ball B = B(0, r) small enough, for any open Zp-latticeH ⊂ B
f (x + H) = f (x) + f ′(x) · H.
RemarkThis allows more models of precision, like
(x , y) = (1 + O(p10), 1 + O(p)).
RemarkOur framework can be extended to (complete) ultrametric K -vectorspaces (e.g. being Fp((X ))n, Q((X ))m, R((ε))s).
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Division and Slope Factorization of p-Adic PolynomialsDivision 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
+∞∑k=2
1k! f (k)(x) · Hk ⊂ f ′(x) · H.
This can be determined with Newton-polygon techniques.
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Division and Slope Factorization of p-Adic PolynomialsDivision 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
+∞∑k=2
1k! f (k)(x) · Hk ⊂ f ′(x) · H.
This can be determined with Newton-polygon techniques.
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Division and Slope Factorization of p-Adic PolynomialsDivision 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
+∞∑k=2
1k! f (k)(x) · Hk ⊂ f ′(x) · H.
This can be determined with Newton-polygon techniques.
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Division and Slope Factorization of p-Adic PolynomialsDivision 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
+∞∑k=2
1k! f (k)(x) · Hk ⊂ f ′(x) · H.
This can be determined with Newton-polygon techniques.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Study of the division
Table of contents
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Study of the division
Handling the precision
Lattice precisionTo each polynomial, we attach a Zp-lattice (given by a basis of thislattice).
Lattice and divisionA and B are known with precision lattice HA and HB . Then (HQ , HR) aregiven 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Study of the division
Handling the precision
Lattice precisionTo each polynomial, we attach a Zp-lattice (given by a basis of thislattice).
Lattice and divisionA and B are known with precision lattice HA and HB . Then (HQ , HR) aregiven 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Study of the division
Handling the precision
Lattice precisionTo each polynomial, we attach a Zp-lattice (given by a basis of thislattice).
Lattice and divisionA and B are known with precision lattice HA and HB . Then (HQ , HR) aregiven 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Study of the division
Handling the precision
Lattice precisionTo each polynomial, we attach a Zp-lattice (given by a basis of thislattice).
Lattice and divisionA and B are known with precision lattice HA and HB . Then (HQ , HR) aregiven 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Study of the division
Handling the precision
Lattice precisionTo each polynomial, we attach a Zp-lattice (given by a basis of thislattice).
Lattice and divisionA and B are known with precision lattice HA and HB . Then (HQ , HR) aregiven 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.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Table of contents
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Lattice precision and modular multiplication
Composed derivativesIt is easy to handle in an optimal way the modular multiplication byapplying:
For A × B,δ(A × B) = AδB + BδA.
For remainder R in A = BQ + R,
δ(R) = δA − QδB mod B.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Lattice precision and modular multiplication
Composed derivativesIt is easy to handle in an optimal way the modular multiplication byapplying:
For A × B,
δ(A × B) = AδB + BδA.
For remainder R in A = BQ + R,
δ(R) = δA − QδB mod B.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Lattice precision and modular multiplication
Composed derivativesIt is easy to handle in an optimal way the modular multiplication byapplying:
For A × B,δ(A × B) = AδB + BδA.
For remainder R in A = BQ + R,
δ(R) = δA − QδB mod B.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Lattice precision and modular multiplication
Composed derivativesIt is easy to handle in an optimal way the modular multiplication byapplying:
For A × B,δ(A × B) = AδB + BδA.
For remainder R in A = BQ + R,
δ(R) = δA − QδB mod B.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Lattice precision and modular multiplication
Composed derivativesIt is easy to handle in an optimal way the modular multiplication byapplying:
For A × B,δ(A × B) = AδB + BδA.
For remainder R in A = BQ + R,
δ(R) = δA − QδB mod B.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Toward numerical understanding
Displaying precision
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Toward numerical understanding
Displaying precisionLet H ⊂ Qn
p be a lattice.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Toward numerical understanding
Displaying precisionLet H ⊂ Qn
p be a lattice.
H0 = π1(H) ⊕ · · · ⊕ πn(H)
provides the best precision oncoordinates.It is the smallest diagonal latticecontaining H.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Toward numerical understanding
Displaying precisionLet H ⊂ Qn
p be a lattice.
H0 = π1(H) ⊕ · · · ⊕ πn(H)
provides the best precision oncoordinates.It is the smallest diagonal latticecontaining H.
Diffused digitsThe number of diffused digits of precisionof H is the length of H0/H.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
More on diffused digits
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
More on diffused digits
Diffused digitsThe number of diffused digits of precisionof H is the length of H0/H.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
More on diffused digits
Diffused digitsThe number of diffused digits of precisionof H is the length of H0/H.
Another definitionHere, the number of diffused digits is:
− logp (|H0/H|) .
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Diffused digits: example
Diffused digits: the latticeLet
H =⟨ 1
pp2
,
pp2
2p4
,
2pp2
2p3
⟩.
Then
H0 =
1p
p2
.
Number of diffused digits
The SNF of H0/H is:
1p
p
. Hence 2 diffused digits.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Diffused digits: example
Diffused digits: the latticeLet
H =⟨ 1
pp2
,
pp2
2p4
,
2pp2
2p3
⟩.
Then
H0 =
1p
p2
.
Number of diffused digits
The SNF of H0/H is:
1p
p
. Hence 2 diffused digits.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Diffused digits: example
Diffused digits: the latticeLet
H =⟨ 1
pp2
,
pp2
2p4
,
2pp2
2p3
⟩.
Then
H0 =
1p
p2
.
Number of diffused digits
The SNF of H0/H is:
1p
p
.
Hence 2 diffused digits.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Diffused digits: example
Diffused digits: the latticeLet
H =⟨ 1
pp2
,
pp2
2p4
,
2pp2
2p3
⟩.
Then
H0 =
1p
p2
.
Number of diffused digits
The SNF of H0/H is:
1p
p
. Hence 2 diffused digits.
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Comparison: ∏ni=1 Ai mod M
Modulus M nGain of precision
Jagged Lattice(not dif. + dif.)
X 5 + X 2 + 1(Irred. mod 2)
10 0.2 0.2 + 0.050 4.2 4.2 + 0.0100 11.2 11.2 + 0.0
X 5 + 1(Sep. mod 2)
10 0.4 0.9 + 6.050 5.6 11.1 + 42.0100 13.6 27.0 + 87.0
X 5 + 2(Eisenstein)
10 6.2 6.2 + 0.050 44.0 44.0 + 0.0100 92.5 92.5 + 0.0
(X + 1)5 + 2(Shift Eisenstein)
10 0.6 4.7 + 1.450 7.1 42.6 + 1.4100 15.1 91.8 + 1.4
X 5 + X + 2(Two slopes)
10 1.7 7.9 + 9.850 8.1 70.7 + 59.8100 16.1 152.6 + 125.9
Figure: Precision for modular multiplication
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilities
Input: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
no diffused digits
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: two possibilitiesInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
×Ai
Zp[X ] Zp[X ]
diffused digits
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 0
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 1
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 2
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 3
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 3
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 4
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 5
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 5
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 10
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Division and Slope Factorization of p-Adic PolynomialsDivision and Differential Precision
Modular Multiplication
Qualitative understanding: long-termInput: P, A1, . . . , An ∈ Zp[X ]d known at precision O(pN)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 latticei = 100
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Table of contents
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Newton polygon of a polynomial
DefinitionLet f (X ) = a0 + · · · + anX n ∈ 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 hullof U.By lower convex hull, we mean the points of the convex hull of U belowthe straight line from (0, (a0)) to (n, v(an)).
PropositionLet Newtf be the (point-wise) biggest convex mapping below U. Thenthe graph of Newtf is (the lower frontier of) NP(f ).
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Newton polygon of a polynomial
DefinitionLet f (X ) = a0 + · · · + anX n ∈ 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 hullof U.By lower convex hull, we mean the points of the convex hull of U belowthe straight line from (0, (a0)) to (n, v(an)).
PropositionLet Newtf be the (point-wise) biggest convex mapping below U. Thenthe graph of Newtf is (the lower frontier of) NP(f ).
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Newton polygon of a polynomial
DefinitionLet f (X ) = a0 + · · · + anX n ∈ 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 hullof U.
By lower convex hull, we mean the points of the convex hull of U belowthe straight line from (0, (a0)) to (n, v(an)).
PropositionLet Newtf be the (point-wise) biggest convex mapping below U. Thenthe graph of Newtf is (the lower frontier of) NP(f ).
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Newton polygon of a polynomial
DefinitionLet f (X ) = a0 + · · · + anX n ∈ 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 hullof U.By lower convex hull, we mean the points of the convex hull of U belowthe straight line from (0, (a0)) to (n, v(an)).
PropositionLet Newtf be the (point-wise) biggest convex mapping below U. Thenthe graph of Newtf is (the lower frontier of) NP(f ).
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Newton polygon of a polynomial
DefinitionLet f (X ) = a0 + · · · + anX n ∈ 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 hullof U.By lower convex hull, we mean the points of the convex hull of U belowthe straight line from (0, (a0)) to (n, v(an)).
PropositionLet Newtf be the (point-wise) biggest convex mapping below U. Thenthe graph of Newtf is (the lower frontier of) NP(f ).
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3
NP(P)
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Vocabulary
DefinitionA slope of the Newton polygon of f is an element of Newt ′
f ([0, n]).
DefinitionIf λ is a slope of Newtf , we call segment of slope λ of Newtf the set{(x , Newtf (x))⧸Newt ′
f (x) = λ}.
DefinitionThe length of this slope is the length of its projection on the x -axis.
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Fundamental theorem of Newton polygons
Theoremf 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, isthe length of the segment of slope −λ of Newtf .
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Fundamental theorem of Newton polygons
Theoremf 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, isthe length of the segment of slope −λ of Newtf .
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3
NP(P)
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3
NP(P)
One slope for 0 of length 2, one slope 1/2 of length 2.
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3
NP(P)
One slope for 0 of length 2, one slope 1/2 of length 2.
Two roots of valuation 0, two of valuation −1/2.
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Basic operations
Proposition (Addition)If f and g are two polynomials, then the Newton polygon of f + g can belower-bounded by taking the lower convex hull for the vertices of Newtfand Newtg .
Proposition (Multiplicativity)If f and g are two polynomials, then the Newton polygon of fg has forslopes that of f and g, with length the sum of that of f and g.
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Basic operations
Proposition (Addition)If f and g are two polynomials, then the Newton polygon of f + g can belower-bounded by taking the lower convex hull for the vertices of Newtfand Newtg .
Proposition (Multiplicativity)If f and g are two polynomials, then the Newton polygon of fg has forslopes that of f and g, with length the sum of that of f and g.
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Some remarks
PropositionIf P ∈ Qp[X ] is irreducible, then all its roots have the same valuation.Hence, NewtP has only one slope.
RemarkThe converse is false. For instance, (X − 1)(X − 2) over Q5.
CorollaryIf NewtP has more than one slope, P is not irreducible.
RemarkThere are good irreducibility criterion based on testing whether one slopecan be obtained by multiplication (namely, Dumas and Eisenstein).
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Some remarks
PropositionIf P ∈ Qp[X ] is irreducible, then all its roots have the same valuation.Hence, NewtP has only one slope.
RemarkThe converse is false. For instance, (X − 1)(X − 2) over Q5.
CorollaryIf NewtP has more than one slope, P is not irreducible.
RemarkThere are good irreducibility criterion based on testing whether one slopecan be obtained by multiplication (namely, Dumas and Eisenstein).
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Basics
Some remarks
PropositionIf P ∈ Qp[X ] is irreducible, then all its roots have the same valuation.Hence, NewtP has only one slope.
RemarkThe converse is false. For instance, (X − 1)(X − 2) over Q5.
CorollaryIf NewtP has more than one slope, P is not irreducible.
RemarkThere are good irreducibility criterion based on testing whether one slopecan be obtained by multiplication (namely, Dumas and Eisenstein).
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Table of contents
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
What is the Newton polygon of the remainder in the division of Aby B (in Qp[X ])?What is the Newton polygon of the quotient in the division of A byB (in Qp[X ])?
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of puX n by B : puX n = BQ + R
valuation
orderdd − 1
NP(B)
n
u puX n
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of puX n by B : puX n = BQ + R
valuation
orderdd − 1
NP(B)
n
u puX n
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of puX n by B : puX n = BQ + R
valuation
orderdd − 1
NP(B)
n
u puX n
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of puX n by B : puX n = BQ + R
valuation
orderdd − 1
NP(B)
n
u puX n
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of puX n by B : puX n = BQ + R
valuation
orderdd − 1
NP(B)
n
u puX n
NP(R)
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of puX n by B : puX n = BQ + R
valuation
orderdd − 1
NP(B)
n
u puX n
NP(R)
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of puX n by B : puX n = BQ + R
valuation
orderdd − 1
NP(B)
n
u puX n
NP(R)
n − d
NP(Q)
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of A by B : A = BQ + R
valuation
orderdd − 1
NP(B)NP(A)
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of A by B : A = BQ + R
valuation
orderdd − 1
NP(B)NP(A)
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Euclidean division
Euclidean division and Newton polygon
Lemma (Division lemma)
Division of A by B : A = BQ + R
valuation
orderdd − 1
NP(B)NP(A)NP(R) NP(Q)
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Precision: Return on Modular Multiplication
Table of contents
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Precision: Return on Modular Multiplication
Handling the precision
Lattice precisionA and B are known with precision lattice HA and HB . Then (HQ , HR) aregiven by the Euclidean division of HA − QHB by B.
Newton precisionFor A = BQ + R with A known with precision-polygon φ, we can applythe previous construction to φ divided by B to obtain the precision on Qand R.
RemarkThis proved to be useful to handle precision for the computation of thecharacteristic polynomial.
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Precision: Return on Modular Multiplication
Handling the precision
Lattice precisionA and B are known with precision lattice HA and HB . Then (HQ , HR) aregiven by the Euclidean division of HA − QHB by B.
Newton precisionFor A = BQ + R with A known with precision-polygon φ, we can applythe previous construction to φ divided by B to obtain the precision on Qand R.
RemarkThis proved to be useful to handle precision for the computation of thecharacteristic polynomial.
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Precision: Return on Modular Multiplication
Handling the precision
Lattice precisionA and B are known with precision lattice HA and HB . Then (HQ , HR) aregiven by the Euclidean division of HA − QHB by B.
Newton precisionFor A = BQ + R with A known with precision-polygon φ, we can applythe previous construction to φ divided by B to obtain the precision on Qand R.
RemarkThis proved to be useful to handle precision for the computation of thecharacteristic polynomial.
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Division and Slope Factorization of p-Adic PolynomialsNewton Polygons
Precision: Return on Modular Multiplication
Comparison: ∏ni=1 Ai mod M
Modulus M nGain of precision
Jagged Newton Lattice(not dif. + dif.)
X 5 + X 2 + 1(Irred. mod 2)
10 0.2 0.2 0.2 + 0.050 4.2 4.2 4.2 + 0.0100 11.2 11.2 11.2 + 0.0
X 5 + 1(Sep. mod 2)
10 0.4 0.4 0.9 + 6.050 5.6 5.6 11.1 + 42.0100 13.6 13.6 27.0 + 87.0
X 5 + 2(Eisenstein)
10 6.2 6.2 6.2 + 0.050 44.0 44.0 44.0 + 0.0100 92.5 92.5 92.5 + 0.0
(X + 1)5 + 2(Shift Eisenstein)
10 0.6 0.6 4.7 + 1.450 7.1 7.1 42.6 + 1.4100 15.1 15.1 91.8 + 1.4
X 5 + X + 2(Two slopes)
10 1.7 1.7 7.9 + 9.850 8.1 8.1 70.7 + 59.8100 16.1 16.1 152.6 + 125.9
Figure: Precision for modular multiplication
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Table of contents
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Factoring respecting slopes
TheoremLet f ∈ Qp[X ]. Then,
We can write f =∏
i fi .The fi ’s are all of one slope.They have respectively different slopes.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Factoring respecting slopes
TheoremLet f ∈ Qp[X ]. Then,
We can write f =∏
i fi .
The fi ’s are all of one slope.They have respectively different slopes.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Factoring respecting slopes
TheoremLet f ∈ Qp[X ]. Then,
We can write f =∏
i fi .The fi ’s are all of one slope.
They have respectively different slopes.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Factoring respecting slopes
TheoremLet f ∈ Qp[X ]. Then,
We can write f =∏
i fi .The fi ’s are all of one slope.They have respectively different slopes.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3.
NP(P)
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3.
NP(P)
NP(f1)
NP(f2)
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3.
NP(P)
NP(f1)
NP(f2)
P = (2 + 3X + X 2) × (1 + 3X 2)
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
例
A Newton polygon
1
2
0 1 2 3 4
P = 2 + 3X + 7X 2 + 9X 3 + 3X 4, over Q3.
NP(P)
NP(f1)
NP(f2)
P = (2 + 3X + X 2) × (1 + 3X 2)
Remark: 2 + 3X + X 2 = (1 + X )(1 + 2X ).
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
A Newton iteration
The iterationAlready found in Polynomial root finding over local rings and applicationto error correcting codes by Berthomieu, Lecerf, Quintin:
Ai+1 := Ai + (ViP mod Ai)Bi+1 := P \quo Ai+1
Vi+1 := (2Vi − V 2i Bi+1) mod Ai+1
The resultAi , Bi , Vi converge quadratically to A, B, V such that AB = P, V is theinverse of B modulo A and A, B have the desired Newton polygons.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Ideas on the proof
About the proofWe monitor Ri = Ai+1 − Ai and Si = P mod Ai .
We prove that both NP(Ri) and NP(Si) goes to infinity.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step.
κ
0 dd−1 d+1
A0
NF (P)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step. Euclidean division.
κ
0 dd−1 d+1
A0
NF (P)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step. Euclidean division.
κ
0 dd−1 d+1
A0
NF (P)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step. Euclidean division.
κ
0 dd−1 d+1
A0
NF (P)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step. Euclidean division.
κ
0 dd−1 d+1
A0
NF (P)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step. Euclidean division.
κ
0 dd−1 d+1
A0
NF (P)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step.
κ
0 dd−1 d+1
A0
NF (P)
NP(R0)NP(S0)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step.
κ
0 dd−1 d+1
A0
NF (P)
NP(R0)NP(S0)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
First step.
κ
0 dd−1 d+1
A0
NF (P)
NP(R0)NP(S0)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
Second Step.
κ
0 dd−1 d+1
A0
NF (P)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
Second Step.
2κκ
0 dd−1 d+1
A0
NF (P)
NP(R1)NP(S1)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
Second Step.
2κκ
0 dd−1 d+1
A0
NF (P)
NP(R1)NP(S1)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
Third Step.
2κ
4κ
κ
0 dd−1 d+1
A0
NF (P)
NP(R2)NP(S2)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
A Newton scheme
Illustration
Third Step.
2κ
4κ
κ
0 dd−1 d+1
A0
NF (P)
NP(R2)NP(S2)
slope = λ1
slope = λ0φ
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
Table of contents
1 Division and Differential Precisionp-Adic PrecisionStudy of the divisionModular Multiplication
2 Newton PolygonsBasicsEuclidean divisionPrecision: Return on Modular Multiplication
3 Slope factorizationA Newton schemeApplying differential precision
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
What about precision?
SettingLet FA : P 7→ A(1) be the application such that P = A(1)B(1), withA(1), B(1) corresponding to the slopes before/after the breakpoint d .
DifferentialThe application FA : P 7→ A(1) is of class C1. Its differential at somepoint 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).
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
What about precision?
SettingLet FA : P 7→ A(1) be the application such that P = A(1)B(1), withA(1), B(1) corresponding to the slopes before/after the breakpoint d .
DifferentialThe application FA : P 7→ A(1) is of class C1. Its differential at somepoint 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).
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
Some numerical results: A 7→ AB 7→ A.
Polynomials PrecisionMean gain of precision
Jagged Newton Lattice
dA B absolute −14.5 −14.7 0.0
relative −1.5 −3.6 0.0
dA B absolute 0.0 0.0 0.0
relative −0.3 −1.2 0.0
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
To sum up
On p-adic precision
Step-by-step analysis: as a first step. Can show differentiability andnaïve 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 computationsDiffused digits for modular multiplication.Newton iteration for slope factorisation.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
To sum up
On p-adic precisionStep-by-step analysis: as a first step. Can show differentiability andnaïve 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 computationsDiffused digits for modular multiplication.Newton iteration for slope factorisation.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
To sum up
On p-adic precisionStep-by-step analysis: as a first step. Can show differentiability andnaïve 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 computationsDiffused digits for modular multiplication.Newton iteration for slope factorisation.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
To sum up
On p-adic precisionStep-by-step analysis: as a first step. Can show differentiability andnaïve 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 computationsDiffused digits for modular multiplication.Newton iteration for slope factorisation.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
To sum up
On p-adic precisionStep-by-step analysis: as a first step. Can show differentiability andnaïve 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.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
To sum up
On p-adic precisionStep-by-step analysis: as a first step. Can show differentiability andnaïve 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 computationsDiffused digits for modular multiplication.
Newton iteration for slope factorisation.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
To sum up
On p-adic precisionStep-by-step analysis: as a first step. Can show differentiability andnaïve 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 computationsDiffused digits for modular multiplication.Newton iteration for slope factorisation.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
References
Initial article
Xavier Caruso, David Roe and Tristan Vaccon Tracking p-adicprecision, ANTS XI, 2014.
Linear Algebra
Xavier Caruso, David Roe and Tristan Vaccon p-adic stability inlinear algebra, ISSAC 2015.
Polynomial Computations
Xavier Caruso, David Roe and Tristan Vaccon Division andSlope Factorization of p-Adic Polynomials, ISSAC 2016.
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Division and Slope Factorization of p-Adic PolynomialsSlope factorization
Applying differential precision
Thank you for your attentionThanks
x + B
f
B
f ′(x)
f (x) + f ′(x) · B
f ′(x) · B
f̃
x
x + O(pN′) y + O(pM′) ⊂ f (x) + O(pN)
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