Solubility of a resultant equation and applications

TL;DR

This paper uses the large sieve to estimate the density of quadratic polynomials with certain resultants and explores their roots modulo primes, with applications to class groups of quadratic fields.

Contribution

It introduces a novel application of the large sieve to quadratic polynomials and connects it to class group size estimates using recent number theory results.

Findings

Density estimates for quadratic polynomials with specific resultants

Existence of primes sharing roots with given polynomials

Application to average class group sizes in quadratic fields

Abstract

The large sieve is used to estimate the density of integral quadratic polynomials $Q$, such that there exists an odd degree integral polynomial which has resultant $\pm 1$ with $Q$. Given a monic integral polynomial $R$ of odd degree, this is used to show that for almost all integral quadratic polynomials $Q$, there exists a prime $p$ such that $Q$ and $R$ share a common root in the algebraic closure of the finite field with $p$ elements. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given.