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This is a Quantitative Study study.

October 23, 2025Open Access

Solubility of a resultant equation and applications

Key Points

Solubility of resultant equations informs polynomial density estimations with quadratic polynomials.
Key finding reveals a common root exists for quadratic polynomials and monic polynomials for primes.
Assessment using a large sieve approach provides insights into odd degree polynomial behavior.
Implications point to applications in understanding quadratic number fields and their average class group sizes.

Abstract

The large sieve is used to estimate the density of quadratic polynomials Q∈ℤx, such that there exists an odd degree polynomial defined over ℤ which has resultant ±1 with Q. Given a monic polynomial R∈ℤx of odd degree, this is used to show that for almost all quadratic polynomials Q∈ℤx, 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.

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