P-adic Arithmetic Research Articles

On efficient computation of sums of characters on the basis of A. G. Postnikov methods

An efficient p-adic method and the structure of an algorithm for computing the sums of characters of finite abelian groups are presented. The method and algorithm are based on the A.G. Postnikov summation method of characters modulo a prime power and its developments. A brief survey of the theory of characters of finite abelian groups, p-adic arithmetic and analysis is presented. Questions of the efficiency of p-adic methods are discussed. Moreover, we present results of computation of other types of sums of characters (Kloosterman sums), which are connecting with Artin-Schreier coverings over prime finite fields. The corresponding method and algorithm are based on the development of another method by A.G. Postnikov. Examples of computation of sums of characters are given.

P-Adic morphology, biomorphic structures and number theory

The purpose of this article is to establish a link between number theory and morphology and to justify the possibility of using p-adic arithmetic to model biological forms. The work consists of two parts. In the first part we develop some mathematical technique which is based on the theory of locally compact abelian groups and on the concept of p-adic spectrum of series, sequences and infinite products. In the second part, this technique is applied to some classical series and infinite products in order to reveal their new properties and to construct some biomorphic structures, which arise as continuous images of systems of 2-adic balls (or spheres) when they are mapped into the complex plane. The article also introduces the concept of Ω-classes of natural numbers, which are the 2-adic analogue of the residual classes in modular arithmetic. Their connection with the process of discrete 2-adic diffusion is shown. It is shown that Ω-classes produce biomorphic forms with bilateral symmetry and also induce on the set of natural numbers a countable set of multidimensional symmetries. The paper also introduces the concept of the cellular structure of the set of natural numbers and describes a method for constructing from these cells 2-adic organisms based on the metaphor of biological morphogenesis.

Solving p-adic polynomial systems via iterative eigenvector algorithms

ABSTRACT We describe an implementation of a polynomial system solver to compute the approximate solutions of a 0-dimensional polynomial system with finite precision p-adic arithmetic. We also describe an improvement to an algorithm of Caruso, Roe, and Vaccon for calculating the eigenvalues and eigenvectors of a p-adic matrix.

Painlevé VI equations in p-adic time

Using the description of Painleve VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs, we translate it into the realm of Buium’s p-adic Arithmetic Differential Equations, where the role of derivative is played by a version of Fermat quotient.

Parallel Implementation of Exact Matrix Computation Using Multiple P-adic Arithmetic

A P-adic Exact Scientific Computational Library (ESCL) for rational matrix operations has been developed over the past few years. The effort has been focusing on converting all rational number operations to integer calculation, and fully taking advantage of the fast integer multiplication of modern computer architectures. In this paper, we report our progress on parallel implementation of P-adic arithmetic by means of a multiple modulus rational system related to the Chinese remainder theorem. Experimental results are given to illustrate computational efficiency.

A notion of diaphony based on p-adic arithmetic

A notion of diaphony based on p-adic arithmetic

O( n3) noniterative heuristic algorithm for linear programs with error-free implementation

We present an O( n 3) noniterative heuristic algorithm and describe an implementation of p-adic arithmetic to solve the linear programming problems error-free. An optimality test as well as numerical examples depict the scope of such an algorithm. In the few cases where it did not lead to the optimal solution, it did provide a nonnegative solution of the system A x = b , which could be used as the basic feasible solution for any one of the widely used simplex methods, say, the revised simplex algorithm without the need to include artificial variables in A x = b for consistency check. Thus the algorithm could be at least a pre-processor for any simplex algorithm and avoids the enhancement of dimensions of the constraint matrix A and consequent additional space, computational complexities, and errors.

Solving linear programming problems exactly

A linear programming problem in an inequality form having a bounded solution is solved error-free using an algorithm that sorts the inequalities, removes the redundant ones, and uses the p-adic arithmetic.

On an Efficient Algorithm for Big Rational Number Computations by Parallel p-adics

This paper presents an algorithm for evaluating an arithmetic expression over "big" rational numbers. The method exploits p-adic arithmetic and parallelism to achieve efficiency. Roughly, the algorithm begins by mapping the input rational numbers to the related p-adic codes for several prime bases, then the corresponding expression is evaluated over these codes and finally the rational output is recovered from the resulting codes. All these three steps are parallelized. An efficient algorithm to compute the recovery step is proposed. Let p be the maximum of the prime bases for the codes, let r be their truncation order and let k be the number of processors. The asymptotic bit-wise complexity of the proposed recovery step is O( r ( k log p) log 23 . The previous known bound was O( r 2( k log p) log 23 ). Let n be the number of operations in the given expression: the asymptotic bitwise complexity of the proposed algorithm for evaluating arithmetic expressions is O( n r 2 log p).

The binomial theorem and the extended Pascal's triangle

With the binomial coefficients ( k n ) being defined for all integers n,k, several forms of the binomial theorem, valid for all n, are provided. This is set up using algebraic means on infinitely long rows of numbers. When diverging series result, they are evaluated modulo an infinite number. This modular arithmetic, related to p-adic arithmetic, thus provides a new way to interpret diverging series.

Comments on "Matrix Processors Using p-Adic Arithmetic for Exact Linear Computations"

The addition and multiplication algorithms of two Hensel codes were presented in two earlier papers [1], [2] on p-adic arithmetic. It is shown here that these algorithms do not always generate a correct result having the same code word length as the two operands and the correct algorithms are given.

P-adic transforms and digital convolution

Krishnamurthy discussed the idea of using p-adic arithmetic to perform exact rational calculations. The possibility of using Krishnamurthy's method to evaluate number-theoretic transforms has been raised. Unfortunately, various inaccurate statements were made about the sequence length and the dynamic range for which these methods are applicable. A proper analysis of the problem shows that the proposed method is in fact rather costly to implement.

Error Free Computation: A Direct Method to Convert Finite-Segment p-Adic Numbers into Rational Numbers

Elements of finite-segment p-adic arithmetic are briefly recalled, and a new direct method to convert finite-segment p-adic numbers into rational numbers is described.

A New Representation of the Rational Numbers for Fast Easy Arithmetic

A novel system for representing the rational numbers based on Hensel’s p-adic arithmetic is proposed. The new scheme uses a compact variable-length encoding that may be viewed as a generalization of radix complement notation. It allows exact arithmetic, and approximate arithmetic under programmer control. It is superior to existing coding methods because the arithmetic operations take particularly simple, consistent forms. These attributes make the new number representation attractive for use in computer hardware.

Matrix Processors Using p-adic Arithmetic for Exact Linear Computations

A unique code (called Hensel's code) is derived for a rational number by truncating its infinite p-adic expansion. The four basic arithmetic algorithms for these codes are described and their application to rational matrix computations is demonstrated by solving a system of linear equations exactly, using the Gaussian elimination procedure.