Alec Shute

Polynomials represented by norm forms via the beta sieve

Given a polynomial f and a number field K, under what circumstances does there exist a rational number t such that f(t) is the norm of a nonzero element of K? In order to address this question, local to global principles are studied for the affine equation given by the polynomial equal to the norm form corresponding to K. This problem has received much attention over the years, but our understanding is largely limited to polynomials and number fields of low degree. A notable exception is the work of Browning and Matthiesen from 2017, which treats polynomials that are products of arbitrarily many linear factors. In this talk, I will present work in progress where I establish the Hasse principle for products of arbitrarily many linear, quadratic or cubic factors, under certain assumptions on K. My approach makes use of the beta sieve first introduced by Rosser and Iwaniec.