Dutch Analytic Number Theory Symposium

Friday, October 24 in Minnaert 2.01 at Utrecht University

Schedule

13:00 - 13:30   Peter Koymans (Universiteit Utrecht)

13:35 - 14:05   Sebastián Carrillo Santana (Universiteit Utrecht)

14:10 - 14:40   David Hokken (Universiteit Utrecht)

14:40 - 15:00   Coffee break

15:00 - 16:00   Vandita Patel (University of Manchester)

16:00 - 16:20   Coffee Break

16:20 - 17:20   Julian Lyczak (Universiteit Antwerpen)

18:30 - 20:30   Dinner*

*If you would like to join for the dinner, please e-mail L.Thompson@uu.nl

Abstracts

Peter Koymans

Title: Value sets of binary forms

Abstract: Given a binary form F with integer coefficients, we define its value set Val(F) to be the set F(x, y) for integers x, y. For two binary forms F and G, what can we say if Val(F) = Val(G)? The goal of this talk is to show how one may give a complete answer to this question. This is joint work with Etienne Fouvry.

Sebastián Carrillo Santana

Title: Powerfree integers and Fourier bounds

Abstract: We develop a general approach for showing when a set of integers A has infinitely many k^{th} powerfree numbers without relying on equidistribution estimates for A. In particular, we show that if the Fourier transform of A satisfies certain moment bounds, and is also "decreasing" in some sense, then A contains infinitely many k^{th} powerfree numbers. We then use this method to show that there are infinitely many cubefree palindromes in base b>=1100, and in the process we obtain new L^1 bounds for the Fourier transform of the set of palindromes. We also show that there are infinitely many squarefree integers such that its reverse is also squarefree in any base b>=2. Moreover, we show that there are infinitely many squarefree integers with a missing digit in base b>= 5, and infinitely many such cubefree integers in base b>=3. 

David Hokken
Title: Irreducibility of random reciprocal polynomials
Abstract: Given an integer H > 1 and random variables a_0, a_1, … taking values in {1, 2, …, H}, define the random monic polynomial f(x) = x^n + a_{n-1} x^{n-1} + … + a_0. It is a long-held belief, going back in various forms to Hilbert, Van der Waerden, and Odlyzko--Poonen, that with high probability f is irreducible and has a large Galois group over the rationals. Recent advances on this topic — both in the large box model (the fixed n, large H limit) and in the restricted coefficient model (the fixed H, large n limit) — mostly assume the independence of the a_j. In this talk, I will discuss joint work with Dimitris Koukoulopoulos in which we study the restricted coefficient model for random reciprocal polynomials of even degree. These satisfy a_{n-j} = a_j for all j. In spite of the dependence between the coefficients, we prove that, with high probability, f is still irreducible and has large Galois group over Q.

Vandita Patel

Title: Power values of power sums

Abstract: we discuss key results and milestones achieved while studying certain families of Diophantine equations as well as touching on open problems. We note that this is an overview of a large body of work involving multiple collaborators, including; A. Argáez-García (UADY), M. Bennett (UBC), N. Coppola (Padova), M. Curcó-Iranzo (Utrecht), S. Siksek (Warwick), M. Khawaja (Warwick) and Ö. Ülkem (Academia Sinica).

Julian Lyczak

Title: Counting quadratic points on Fano varieties

Abstract: I will present a general framework for counting quadratic points of bounded height on a Fano variety X. If X is a surface, the outcome of this counting problem is predicted by the Manin-Peyre conjecture for the symmetric square of X. I will explain how the framework can be used to verify this conjecture for the infinite family of symmetric squares of non-split quadric surfaces. This talk is based on joint work with Francesca Balestrieri, Kevin Destagnol, Jennifer Park and Nick Rome.