Schinzel Hypothesis on average and rational points

Abstract

We prove the existence version of Schinzel's Hypothesis (H) for 100% of integer polynomials P 1 , . . . , P n of fixed degrees, when ordered by the size of coefficients. We deduce that a positive proportion of diagonal conic bundles over Q with any given number of degenerate fibres have a rational point, and obtain similar results for generalised Châtelet equations.

Average Bateman--Horn for Kummer polynomials

“…It should be possible to extend the methods of this paper to handle composite r and this will be the subject of future work. Recently, Sofos and Skorobogatov [30] have independently investigated an average form of the Bateman-Horn conjecture and shown that, if we order polynomials by the size of their coefficients, then 100 % of them satisfy the Bateman-Horn conjecture. However, their work and their techniques cannot be used to provide insight into thin families of polynomials such as the ones that we consider in this paper (more discussion on this point will follow in the next subsection).…”

“…However, their work and their techniques cannot be used to provide insight into thin families of polynomials such as the ones that we consider in this paper (more discussion on this point will follow in the next subsection). The more arithmetic motivation for our work (as well as that of [30]) is that many results about the qualitative behaviour of rational points on varieties with a fibration structure are known to hold under the so-called Schinzel's hypothesis, which is a special case of the Bateman-Horn conjecture. Indeed the first such example of this is the use of Dirichlet's theorem on primes in arithmetic progressions in the proof of the Hasse-Minkowski theorem.…”

“…The proof of this theorem consists in establishing, under the relevant conditions on a, that some variants of the Hasse principle and of the integral Hasse principle hold for X a,r,k . As our main tool, we use the analytic input together with a modification of standard fibration method arguments, similar to the one used in [30]. This tight interplay between geometry and analysis is in the spirit of the results of [5] and [23].…”

“…We also note that, if we were to follow the alternative approach used in [30], we would have to open the square in the left-hand side of (1.1) and bound a sum of the form…”

The Hilbert-Schinzel specialization property

“…6 Passing from "at least one" to "infinitely many" prime values is not nearly as convenient for the original Schinzel Hypothesis. Indeed, [SS20] establishes asymptotic results showing that "most" irreducible integer polynomials without fixed prime divisors take at least one prime value, whereas the infiniteness assertion is not known for a single non-linear polynomial. 7 As usual, PID stands for Principal Ideal Domain.…”