Generic diagonal conic bundles revisited

TL;DR

This paper proves a stronger form of Schinzel's Hypothesis for almost all polynomial n-tuples with constraints, establishes the triviality of the Brauer group for generic diagonal conic bundles, and provides bounds on the probability of rational points.

Contribution

It advances the understanding of polynomial prime representations under constraints and analyzes the arithmetic of diagonal conic bundles, including their rational points.

Findings

Schinzel's Hypothesis holds for 100% of suitable polynomial n-tuples.

The Brauer group of generic diagonal conic bundles over the projective line is trivial.

Provides explicit lower bounds for the probability of rational points on certain conic bundles.

Abstract

We prove a stronger form of our previous result that Schinzel's Hypothesis holds for $100\%$ of $n$-tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.