Related papers: Cyclotomy and permutation polynomials of large ind…

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…

We define standardized constructions of finite fields, and standardized generators of (multiplicative) cyclic subgroups in these fields. The motivation is to provide a substitute for Conway polynomials which can be used by various software…

In this paper, we construct several new permutation polynomials over finite fields. First, using the linearized polynomials, we construct the permutation polynomial of the form $\sum_{i=1}^k(L_{i}(x)+\gamma_i)h_i(B(x))$ over ${\bf…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…

We propose an improved algorithm for finding roots of polynomials over finite fields. This makes possible significant speedup of the decoding process of Bose-Chaudhuri-Hocquenghem, Reed-Solomon, and some other error-correcting codes.

We give explicit upper bounds for coefficients of polynomials appearing in Gauss-Kra\"{i}tchik formula for cyclotomic polynomials. We use a certain relation between elementary symmetric polynomials and power sums polynomials.

We give a short proof of polynomial recurrence with large intersection for additive actions of finite-dimensional vector spaces over countable fields on probability spaces, improving upon the known size and structure of the set of strong…

Using cyclotomic classes of order twelve for certain finite fields, we construct an infinite family of almost difference sets and normally regular graphs applying the theory of cyclotomy. We show that in each of these fields neither the…

We derive several existence results concerning cycle types and, more generally, the "mapping behavior" of complete mappings. Our focus is on so-called first-order cyclotomic mappings, which are functions on a finite field $\mathbb{F}_q$…

We introduce the notion of semi-characteristic polynomial for a semi-linear map of a finite- dimensional vector space over a field of characteristic p. This polynomial has some properties in common with the classical characteristic…

Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational…

Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…

In this paper we study a class of dynamical systems generated by iterations of multivariate permutation polynomial systems which lead to polynomial growth of the degrees of these iterations. Using these estimates and the same techniques…

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the…

Four recursive constructions of permutation polynomials over $\gf(q^2)$ with those over $\gf(q)$ are developed and applied to a few famous classes of permutation polynomials. They produce infinitely many new permutation polynomials over…

Many-to-one mappings and permutation polynomials over finite fields have important applications in cryptography and coding theory. In this paper, we study the many-to-one property of a large class of polynomials such as $f(x) = h(a x^q + b…

We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…

Permutation polynomials with coefficients 1 over finite fields attract researchers' interests due to their simple algebraic form. In this paper, we first construct four classes of fractional permutation polynomials over the cyclic subgroup…