Related papers: A structure theorem for finite fields
Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to construct…
In this paper, we improve the finiteness constant for the finiteness principles for $C^m(\mathbb{R}^n,\mathbb{R}^d)$ and $C^{m-1,1}(\mathbb{R}^n,\mathbb{R}^D)$ selection proven by Fefferman, Israel, and the second author and extend the more…
The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of…
We introduce some special presentations on finite groups, that we call "diagonal double Kodaira structures" and whose existence is equivalent to the existence of some special Kodaira fibred surfaces, that we call "diagonal double Kodaira…
End-spaces of infinite graphs naturally generalise the Freudenthal boundary and sit at the interface between graph theory, geometric group theory and topology. Our main result is that every end-space can topologically be represented by a…
Recently Lima and Campello de Souza introduced a new class of rational functions over odd-order finite fields, and explained their potential usefulness in cryptography. We show that these new functions are conjugate to the classical family…
We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive…
A Structure Theorem for Protori is derived for the category of finite-dimensional protori(compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a…
We obtain divisibility conditions on the multiplicative orders of elements of the form $\zeta + \zeta^{-1}$ in a finite field by exploiting a link to the arithmetic of real quadratic fields.
We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any subset of a finite field can be decomposed into two disjoint subset $\cU$ and $\cV$ of which the additive energy of $\cU$ and $f(\cV)$ are small, for suitably…
We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions. This widens the class of functions that can be modeled in type-theory based theorem proving tool to…
We show that descriptive complexity's result extends in High Order Logic to capture the expressivity of Turing Machine which have a finite number of alternation and whose time or space is bounded by a finite tower of exponential. Hence we…
Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…
We restructure and advance the classification theory of finite racks and quandles by employing powerful methods from transformation groups and representation theory, especially Burnside rings. These rings serve as universal receptacles for…
We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches…
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.
In this paper we extend the Tanaka finiteness theorem and inequality for the number of symmetries to arbitrary distributions (differential systems) and provide several applications.
We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.
We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into…
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…