Centre de Recerca Matemàtica, Bellaterra, Barcelona
The theory of polynomials over finite fields is fundamental for the study of finite fields, which in turn plays a central role in many areas of pure and applied mathematics. This is a classical area of mathematics with a rich history, going back to Gauss and Galois. The exciting and challenging problems concerning univariate and multivariate polynomials over finite fields are of intricate algebraic and number theoretic flavour and their study requires deep mathematical and computational tools. The determination and construction of special types of (irreducible, primitive, permutation) univariate and multivariate polynomials, for example, as well as understanding many of their functional and algebraic properties (composition, decomposition, iteration, factorization, size of value sets) are long standing problems in the theory of finite fields. These areas have attracted further attention in recent years due to their applications in cryptography, coding theory, combinatorics, design theory, quasi-Monte Carlo methods, communications.
Although the last decades have witnessed a vast amount of work and meetings on polynomials over finite fields, leading to solutions of some of the fundamental problems, inevitably many new questions and conjectures have emerged, with relevance to new areas of pure and applied mathematics. This workshop aims to provide means to meet these new challenges. To exemplify some of the newly emerged open research directions that will be addressed at this workshop, we can mention functional graphs of polynomials, iterations of polynomials over finite fields and orbit structures. Indeed, novel constructions of non-classical dynamical systems over finite fields and the classification of their anomalous behaviour, which is not present in standard constructions of complex dynamical systems, have led to innovative solutions to problems in cryptography, Monte Carlo methods, biological and physical systems. The combination of problems to be discussed in the proposed workshop are particularly selected not only because they are of great interest in theory and applications, but also because they are closely related and advancement in one may lead to solutions of others.
The research directions proposed for this workshop are closely related to finite fields, number theory, commutative algebra, cryptography, dynamical systems, combinatorics, to mention just a few. Thus, researchers of different backgrounds and in possession of different technical toolboxes sometimes work on similar questions without knowing each other's ideas and results. The goal of the workshop is to establish new collaborations, discuss potential solutions to open problems, formulate new ones, and try to advance the theory with innovative results. We believe that it is time to assess the progress, while proposing future directions of research.
Wen-Ching (Winnie) Li
Joachim von zur Gathen