p-Adic Stability In Linear Algebra

Caruso, Xavier; Roe, David; Vaccon, Tristan

doi:10.1145/2755996.2756655

Cited by 15 publications

(27 citation statements)

References 5 publications

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“…A tricky point when dealing with zealous arithmetic concerns Newton iteration. To illustrate it, let us examine the example of the computation of a square root of c over Q 2 for c = 1 + 2 3 + 2 4 + 2 5 + 2 10 + 2 13 + 2 16 + 2 17 + 2 18 + 2 19…”

Section: Newton Iterationmentioning

confidence: 99%

“…The sum of two lattices, for example, is computed by concatenating the two corresponding Hermite matrices and re-echelonizing. In a similar fashion, one can compute the image of a lattice under a surjective 19 linear mapping f : we apply f to each generator and echelonize the (possibly rectangular) matrix obtained this way.…”

Section:   mentioning

confidence: 99%

“…One can cook up even worse examples. Look for instance at the function f : Z p → Z p defined by: a 0 + a 1 p + · · · + a n p n + · · · → a 0 + a 1 p 2 + · · · + a n p 2n + · · · 19 Surjectivity ensures that the image remains a lattice in the codomain.…”

Section: The Case Of Univariate Functionsmentioning

confidence: 99%

“…Other results concerning more general f and building on the machinery of Newton polygons are available in the literature. We refer the reader to [18] for the case of locally analytic functions and to [19] for the case of solutions of partial differential equations of order 1.…”

Section: The Precision Lemmamentioning

confidence: 99%

“…This sounds quite weird because we were able in any case to compute u 19 using algorithm somos option1. The point is that the formula given the value of u 19 has the shape u 19…”

Section: Example: the Somos 4 Sequencementioning

confidence: 99%

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Computations with p-adic numbers

Caruso¹

2017

Cours du CIRM

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This document contains the notes of a lecture I gave at the "Journées Nationales du Calcul Formel 1 " (JNCF) on January 2017. The aim of the lecture was to discuss low-level algorithmics for p-adic numbers. It is divided into two main parts: first, we present various implementations of p-adic numbers and compare them and second, we introduce a general framework for studying precision issues and apply it in several concrete situations. (French) National Computer Algebra DaysTheory and Arithmetic Geometry. After Diophantine equations, other typical examples come from the study of number fields: we hope deriving interesting information about a number field K by studying carefully all its p-adic incarnations K ⊗ Q Q p . The ramification of K, its Galois properties, etc. can be -and are very often -studied in this manner [69,65]. The class field theory, which provides a precise description of all Abelian extensions 2 of a given number field, is also formulated in this language [66]. The importance of p-adic numbers is so prominent today that there is still nowadays very active research on theories which are dedicated to purely p-adic objects: one can mention for instance the study of p-adic geometry and p-adic cohomologies [6,58], the theory of p-adic differential equations [50], Coleman's theory of padic integration [24], the p-adic Hodge theory [14], the p-adic Langlands correspondence [5], the study of p-adic modular forms [34], p-adic ζ-functions [52] and L-functions [22], etc. The proof of Fermat's last Theorem by Wiles and Taylor [81, 78] is stamped with many of these ideas and developments. Over the last decades, p-adic methods have taken some importance in Symbolic Computation as well. For a long time, p-adic methods have been used for factoring polynomials over Q [56]. More recently, there has been a wide diversification of the use of p-adic numbers for effective computations: Bostan et al. [13] used Newton sums for polynomials over Z p to compute composed products for polynomials over F p ; Gaudry et al. [32] used p-adic lifting methods to generate genus 2 CM hyperelliptic curves; Kedlaya [49], Lauder [54] and many followers used p-adic cohomology to count points on hyperelliptic curves over finite fields; Lercier and Sir vent [57] computed isogenies between elliptic curves over finite fields using p-adic differential equations.The need to build solid foundations to the algorithmics of p-adic numbers has then emerged. This is however not straightforward because a single p-adic number encompasses an infinite amount of information (the infinite sequence of its digits) and then necessarily needs to be truncated in order to fit in the memory of a computer. From this point of view, p-adic numbers behave very similarly to real numbers and the questions that emerge when we are trying to implement p-adic numbers are often the same as the questions arising when dealing with rounding errors in the real setting [62,26,63]. The algorithmic study of p-adic numbers is then located at the frontier between Symbolic Computa...

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Section: Newton Iterationmentioning

confidence: 99%

Section:   mentioning

confidence: 99%

Section: The Case Of Univariate Functionsmentioning

confidence: 99%

Section: The Precision Lemmamentioning

confidence: 99%

“…This sounds quite weird because we were able in any case to compute u 19 using algorithm somos option1. The point is that the formula given the value of u 19 has the shape u 19…”

Section: Example: the Somos 4 Sequencementioning

confidence: 99%

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Computations with p-adic numbers

Caruso¹

2017

Cours du CIRM

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Solving systems of algebraic equations over finite commutative rings and applications

Kamche,

Kalachi

2024

AAECC

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Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A finite chain ring is a finite ring admitting exactly one maximal ideal and every ideal being generated by one element. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residue field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

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Solving p-adic polynomial systems via iterative eigenvector algorithms

Kulkarni

2020

Linear and Multilinear Algebra

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p-Adic Stability In Linear Algebra

Cited by 15 publications

References 5 publications

Computations with p-adic numbers

Computations with p-adic numbers

Solving systems of algebraic equations over finite commutative rings and applications

Solving p-adic polynomial systems via iterative eigenvector algorithms

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