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arXiv · 2507.16138

The structure of the double discriminant

Theresa C. Anderson·Ufuoma V. Asarhasa·Adam Bertelli·Fabian Gundlach·Evan M. O'Dorney
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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.

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Theresa C. Anderson, Ufuoma V. Asarhasa, Adam Bertelli, Fabian Gundlach, Evan M. O'Dorney. 2025-07-22. The structure of the double discriminant. https://arxiv.org/abs/2507.16138

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