Work

Solubility of a resultant equation and applications

Year: 2025

Type: article

Abstract: The large sieve is used to estimate the density of quadratic polynomials Q ∈ ℤ [ x ] , such that there exists an odd degree polynomial defined over ℤ which has resultant ± 1 with Q . Given a monic polynomial R ∈ ℤ [ x ] of odd degree, this is used to show that for almost all quadratic polynomials Q ∈ ℤ [ x ] , there exists a prime p such that Q and R share a common root in 𝔽 ¯ p . 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.

Source: Journal de l’École polytechnique — Mathématiques

DOI: 10.5802/jep.320

Authors: Tim Browning, Stephanie Chan

Institutions: Institute of Science and Technology Austria, University College London

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Austrian Science Fund

Awards: G4495373907, Rational curves via function field analytic number theory