Related papers: On a Class of Permutation Rational Functions Invol…

In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several…

The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to…

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…

The module category of any artin algebra is filtered by the powers of its radical, thus defining an associated graded category. As an extension of the degree of irreducible morphisms, this text introduces the degree of morphisms in the…

In this paper we extend a previous investigation by us regarding an iterative construction of irreducible polynomials over finite fields of odd characteristic. In particular, we show how it is possible to iteratively construct irreducible…

We present new classes of permutation polynomials over finite fields.

The characterization of permutations over finite fields is an important topic in number theory with a long-standing history. This paper presents a systematic investigation of low-degree bivariate polynomial systems $F=(f_1(x,y),f_2(x,y))$…

This paper introduces a robust class of functions from finite words to integers that we call Z-polyregular functions. We show that it admits natural characterizations in terms of logics, Z-rational expressions, Z-rational series and…

We give defining equations for function fields over finite fields with many rational places. They are obtained from composita of quadratic extensions of the rational function field.

We consider chaining multiplicative-inverse operations in finite fields under alternating polynomial bases. When using two distinct polynomial bases to alternate the inverse operation we obtain a partition of $\mathbb F_{p^n}\setminus…

We explore the connection between cyclotomic mapping permutation polynomials and permutation polynomials of the form $x^rf(x^{\frac{q-1}{l}})$ over finite fields. We present a new necessary and a new sufficient condition to verify…

In this paper, we propose linear maps over the space of all polynomials $f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with permutation…

In this paper we present an iterative construction of irreducible polynomials over finite fields based upon repeated applications of transforms induced by endomorphisms of odd prime degree of ordinary elliptic curves.

We construct classes of permutation polynomials over F_{Q^2} by exhibiting classes of low-degree rational functions over F_{Q^2} which induce bijections on the set of (Q+1)-th roots of unity in F_{Q^2}. As a consequence, we prove two…

This work presents an initial analysis of using bijective mappings to extend the Theory of Functional Connections to non-rectangular two-dimensional domains. Specifically, this manuscript proposes three different mappings techniques: a)…

In this paper we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual…

Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results [4,6] on the construction of such invariants by concrete…

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…

This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…