POLYNOMIALS REPRESENTED BY NORM FORMS VIA THE BETA SIEVE

A central question in Arithmetic geometry is to determine for which polynomials f∈Z[t] and which number fields K the Hasse principle holds for the affine equation f(t)=NK/Q(x)≠0 . Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.