Arithmetica İzmir 1

Date: November 16, 2018

Place: Dokuz Eylül University, Mathematics Department, B255

Invited speakers

Alp Bassa, Boğaziçi University

Ayhan Günaydın, Boğaziçi University

Şermin Çam Çelik, Özyeğin University

Doğa Can Sertbaş, Cumhuriyet University

Ayhan Günaydın, Boğaziçi Üniversitesi

Title: Polynomial Exponential Equations

Abstract: Let V be an algebraic set in C^{2n} over complex numbers. We would like to find a generic element of V of the form (x,exp(x)) where exp is the complex exponential map defined on C^n. Of course, we need certain conditions necessary for the existence of such an element. These assumptions will be explained in the beginning of the talk. Then we present a way to change this question into a question about the solutions of polynomial‐exponential equations over the complex numbers where the variables run through the rationals. After that we make a reduction to integers variables. Finally focusing on the case n=1, we explain why such a generic element should exists.
(Joint work with Paola D’Aquino, Antongiulio Fornasiero, and Giuseppina Terzo.)

Alp Bassa, Boğaziçi Üniversitesi

Title: Good Recursive Towers over Prime Fields Exist

Abstract: In the past, various methods have been employed to construct high genus curves over finite fields with many rational points. One such method is by means of explicit recursive towers and will be the emphasis of this talk. The first explicit examples were found by Garcia–Stichtenoth over quadratic finite fields in 1995. Shortly after followed the discovery of good towers over cubic finite fields in 2005 (Bezerra–Garcia-Stichtenoth) and all nonprime finite fields in 2013 (Bassa–Beelen–Garcia–Stichtenoth). The questions of finding good towers over prime fields resisted all attempts for several decades and lead to the common belief that such towers do not exist. In this talk I will try to give an overview of the landscape of explicit recursive towers and present a recently discovered tower over prime fields. This is joint work with Christophe Ritzenthaler.

Doğa Can Sertbaş, Cumhuriyet Üniversitesi

Title: Density results on Egyptian Fractions

Abstract: Any finite sum of distinct unit fractions is called an Egyptian fraction. For some fixed positive natural number k, a restricted Egyptian fraction is also defined as the sum of at most k unit fractions, where the repetition is allowed. These type of fractions have been studied extensively and there are some open problems related to them. In this talk, we introduce some of these problems and mention some arithmetic and analytic properties of restricted Egyptian fractions. In particular, we provide analytic proofs which show that restricted Egyptian fractions are not dense in the interval [0,1] in the sense of height and topology.

Şermin Çam Çelik, Özyeğin Üniversitesi

Title: Special Values of Dirichlet Series

Abstract: In this talk, we will show that real numbers can be strongly approximated by linear combinations of special values of Dirichlet series.