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We find arbitrarily large configurations of irreducible polynomials over finite fields that are separated by low degree polynomials. Our proof adapts an argument of Pintz from the integers, in which he combines the methods of…

Let $n \ge 2$ be an integer. In this note, we show that the {\it oriented} transition matrices over the field $\mathcal R$ of all real numbers (over the finite field $\mathcal Z_2$ of two elements respectively) of all continuous {\it vertex…

We show that there exists a connection between two types of objects: some kind of resultantal varieties over C, from one side, and varieties of twists of the tensor powers of the Carlitz module such that the order of 0 of its L-functions at…

Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb F}_q$ and suppose $W$ and $\widetilde{W}$ are subspaces of $V$. Two linear transformations $T:W\to V$ and $\widetilde{T}:\widetilde{W}\to V$ are said to be similar…

In this paper, we present a linear algebraic approach to the study of permutation polynomials that arise from linear maps over a finite field $\mathbb{F}_{q^2}$. We study a particular class of permutation polynomials over…

If a string is cyclically shifted it will re-appear after a certain number of shifts, which will be called its order. We solve the problem of how many strings exist with a given order. This problem arises in the context of quantum mechanics…

A graph is called (generically) rigid in $\mathbb{R}^d$ if, for any choice of sufficiently generic edge lengths, it can be embedded in $\mathbb{R}^d$ in a finite number of distinct ways, modulo rigid transformations. Here we deal with the…

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. The Carlitz rank of a permutation polynomial is a important measure of complexity of the polynomial. In this paper we find the sharp lower bound for the weight of any permutation…

We derive formulas for $(i)$ the number of toroidal $n\times n$ binary arrays, allowing rotation of rows and/or columns as well as matrix transposition, and $(ii)$ the number of toroidal $n\times n$ binary arrays, allowing rotation and/or…

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is…

We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$,…

Kronecker's Theorem and Rabin's Theorem are fundamental results about computable fields F and the decidability of the set of irreducible polynomials over F. We adapt these theorems to the setting of differential fields K, with constrained…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, and $T$ a positive integer. In this article we find a sharp estimative of the total number of monic irreducible binomials in $\mathbb F_q[x]$ of degree less or equal to $T$, when $T$…

We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem} (are all but finitely many terms of a…

In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affine functions, where $0\neq…

The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and…

Recently Brian Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra $\boxtimes$ by generators and relations, and…

We prove the undecidability of determining whether a Turing machine yields an eventually periodic trajectory. From this, we deduce the undecidability of orbit finiteness in the polynomial dynamical system on infinite tuples of integers.

In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…

Let $n$ be a positive integer, and let $R$ be a (possibly infinite dimensional) finitely presented algebra over a computable field of characteristic zero. We describe an algorithm for deciding (in principle) whether $R$ has at most finitely…