Related papers: Finite field models in arithmetic combinatorics --…
In this work, families of kinks are analytically identified in multifield theories with either polynomial or deformed sine-Gordon-type potentials. The underlying procedure not only allows us to obtain analytical solutions for these models,…
Combining models in appropriate ways to achieve high performance is commonly seen in machine learning fields today. Although a large amount of combinatorial models have been created, little attention is drawn to the commons in different…
In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in…
This survey paper was primarily written as as the support for a course pesented at the JNCF2025: it aims to present some material that illustrates the kind of estimates one can obtain in effective algebraic geometry, for affine polynomial…
We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the…
In 1987 Greene introduced the notion of the finite field analogue of hypergeometric series. In this paper we give a finite field analogue of Appell series and obtain some transformation and reduction formulas. We also establish the…
In the last few years there have been rapid developments in SMT solving for finite fields. These include new decision procedures, new implementations of SMT theory solvers, and new software verifiers that rely on SMT solving for finite…
We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.
We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…
We consider various counting questions for irreducible binomials over finite fields. We use various results from analytic number theory to investigate these questions.
We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by…
This thesis establishes new quantitative records in several problems of incidence geometry and growth. After the necessary background in Chapters 1, 2 and 3, the following results are proven. Chapter 4 gives new results in the incidence…
By [R. Bautista, P. Gabriel, A.V Roiter., L. Salmeron, Representation-finite algebras and multiplicative basis. Invent. Math. 81 (1985) 217-285.], a finite-dimensional algebra having finitely many isoclasses of indecomposable…
This paper studies the structure of finite hyperfields $H$, and finds a subtle pattern in their addition operation. Consider the class $\mathcal{H}$ of all hyperfields with a given multiplicative group on $H^\times = H - \{0\}$ and given…
Associative algebras with involution over a field of zero characteristic are considered. It is proved that in this case for any finitely generated associative algebra with involution there exists a finite dimensional algebra with involution…
Following an idea of Rowland we give a conjectural way to generate increasing sequences of primes using algorithms involving the gcd. These algorithms seem not so useless for searching primes since it appears we found sometime primes much…
We develop a generic computational model that can be used effectively for establishing the existence of winning strategies for concrete finite combinatorial games. Our modelling is (equational) logic-based involving advanced techniques from…
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems…
This paper constructs a combinatorial model for all postcritically finite rational maps arising as the Newton's method of a complex polynomial. This model is used in [LMS] to give a combinatorial classification of postcritically finite…
In various application fields, such as fluid-, cell-, or crowd-simulations, spatial data structures are very important. They answer nearest neighbor queries which are instrumental in performing necessary computations for, e.g., taking the…