The structure of the double discriminant

TL;DR

This paper provides an explicit algebraic factorization of the double discriminant of a polynomial, revealing its structure as a product of a square, a cube, and possibly a linear monomial, with implications in algebraic geometry and number theory.

Contribution

The paper proves an explicit algebraic factorization of the double discriminant, connecting algebraic geometry and number theory insights.

Findings

Double discriminant factors into a square, a cube, and possibly a linear monomial.

The proof is entirely algebraic, avoiding geometric or analytic methods.

The result clarifies the structure of the double discriminant in polynomial algebra.

Abstract

For a polynomial $f(x) = \sum_{i=0}^n a_i x^i$, we study the double discriminant $DD_{n,k} = \operatorname{disc}_{a_k} \operatorname{disc}_x f(x)$. This object has been well studied in algebraic geometry, but has been brought to recent prominence in number theory by its key role in the proof of the Bhargava--van der Waerden theorem. We bridge the knowledge gap for this object by proving an explicit factorization: $DD_{n,k}$ is the product of a square, a cube, and possibly a linear monomial. Our proof is entirely algebraic. We also investigate other aspects of this factorization.