Prime and square-free values of polynomials in moderately many variables

Kevin Destagnol, Efthymios Sofos

Published 2018-01-09Version 1

We study prime values of irreducible polynomials in many variables. The method uses the arguments behind Birch's well-known result regarding the Hasse principle for complete intersections, however, we prove our results in $50\%$ fewer variables than in the Birch setting. As an application, we derive the Hasse principle and weak approximation for pencils of certain varieties in the spirit of work by Colliot-Th\'el\`ene-Sansuc and Harpaz-Skorobogatov-Wittenberg. We also study the problem of square-free values of an integer polynomial with 66.6\% fewer variables than in the Birch setting.