On the Polynomials Congruent Modulo P^a

2002

Journal of Number Theory

Fekete polynomials associate with each prime number p a polynomial with coefficients −1 or 1 except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo p. These polynomials were already considered in the 19th century in relation to the studies of Dirichlet L-functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomials. In particular, we propose two conjectures on the structure of these Galois groups. We also provide some computational evidence toward the validity of these conjectures.

International Journal of Number Theory, 2012

Let p > 5 be a prime. We prove congruences modulo p 3−d for sums of the general form

Acta Arithmetica, 2003

2019

Let p be a prime and let ' 2 Zp[x1, x2, . . . , xp] be a symmetric polynomial, where Zp is the field of p elements. A sequence T in Zp of length p is called a '-zero sequence if '(T ) = 0; a sequence in Zp is called '-zero free if it does not contain any '-zero subsequence. Motivated by the EGZ theorem for the prime p, we define a symmetric polynomial ' 2 Zp[x1, x2, . . . , xp] to be an EGZ polynomial if it satisfies the following two conditions: (i) every sequence in Zp of length 2p 1 contains a 'zero subsequence and (ii) the '-zero free sequences of length 2p 2 are the sequences which contain exactly any two residues where each residue appears p 1 times. For a positive integer k and a prime p, a power-sum polynomial over Zp is defined as sk,p = Pp j=1 x k j . In this paper we answer the question of whether there are EGZ polynomials among the power-sum polynomials. Indeed, sk,p is an EGZ polynomial if and only if gcd(k, p 1) = 1 and these polynomials...

Acta Arithmetica, 2010

2017

We study the solutions of certain congruences in different rings. The congruences include

2019

Since their introduction in 2004, Polynomial Modular Number Systems (PMNS) have become a very interesting tool for implementing cryptosystems relying on modular arithmetic in a secure and efficient way. However, while their implementation is simple, their parameterization is not trivial and relies on a suitable choice of the polynomial on which the PMNS operates. The initial proposals were based on particular binomials and trinomials. But these polynomials do not always provide systems with interesting characteristics such as small digits, fast reduction, etc. In this work, we study a larger family of polynomials that can be exploited to design a safe and efficient PMNS. To do so, we first state a complete existence theorem for PMNS which provides bounds on the size of the digits for a generic polynomial, significantly improving previous bounds. Then, we present classes of suitable polynomials which provide numerous PMNS for safe and efficient arithmetic.

Acta Arithmetica, 2013

Advances in Difference Equations

In recent years, studying degenerate versions of various special polynomials and numbers has attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials, and degenerate type 2 Euler polynomials, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. Regarding those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas, and analogues for the Bernoulli interpretation of powers of the first m positive integers in terms of Bernoulli polynomials. The present study was done by using the bosonic and fermionic p-adic integrals on Z p .