Rational points and prime values of polynomials in moderately many variables

TL;DR

This paper establishes the Hasse principle and weak approximation for certain polynomial-defined varieties, proves Schinzel's hypothesis in fewer variables, and investigates square-free values of polynomials with reduced variable counts.

Contribution

It introduces new methods to prove Schinzel's hypothesis and the Hasse principle for polynomials in fewer variables than classical results, advancing understanding of rational points and prime values.

Findings

Proved the Hasse principle and weak approximation for specific varieties.

Established Schinzel's hypothesis for polynomials with significantly fewer variables.

Analyzed square-free values of polynomials with reduced variable requirements.

Abstract

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. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel's hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch's well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting.