Related papers: Squarefree Values Of Polynomials

In this article we use a method of finding the index of a complex-valued function by determined number of arithmetic operations to describe an algorithm of localization of roots of square-free polynomials. We give an estimation of the…

An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes (see reference [3]). We improve one of the error terms.

We obtain explicit forms of the current best known asymptotic upper bounds for gaps between squarefree integers. In particular we show, for any $x \ge 2$, that every interval of the form $(x, x + 11x^{1/5}\log x]$ contains a squarefree…

In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of the first kind in terms of the (exponential) complete Bell polynomials where the arguments include the generalised harmonic numbers. We also…

We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of…

This text consists of five relatively systematic notes on Gr\"obner bases and free resolutions of modules over solvable polynomial algebras.

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

Properties of 2-adic valuation sequences for general quadratic polynomials with integer coefficients are determined directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at…

In this paper, we consider the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…

We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why…

We investigate a family of permutation polynomials of finite fields of characteristic 2. Through a connection between permutation polynomials and quadratic forms, a general treatment is presented to characterize these permutation…

In this paper we initiate a study on Gauss factorials of polynomials over finite fields, which are the analogues of Gauss factorials of positive integers.

In this paper, we consider the degenerate Carlitz q-Bernoulli numbers and polynomials and we investigate some properties of those polynomials.

In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Levy processes. First, we classify such families that consist of orthogonal polynomials;…

Let $x$ be a positive real number, and $\mathcal{P} \subset [2,\lambda(x)]$ be a set of primes, where $\lambda(x) \in \Omega(x^\varepsilon)$ is a monotone increasing function with $\varepsilon \in (0,1)$. We examine $Q_{\mathcal{P}}(x)$,…

In this paper we introduce a new approach to the concept of multipolynomials and generalize several results of the homogeneous polynomials and symmetric multilinear applications. We also present an abstract approach to the concept of…