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This is a Quantitative Study study.

September 23, 2025Open Access

The structure of the double discriminant

Key Points

The double discriminant is conjectured to be a product of a square, a cube, and possibly a linear monomial.
The factorization behavior of the double discriminant indicates significant integer constant influences.
Proving the conjecture for k=0 establishes a foundational result for further exploration.
Understanding double discriminants enhances comprehension of the van der Waerden--Bhargava theorem.

Abstract

For a polynomial f (x) = ₈=₀ⁿ aᵢ xⁱ, we study the double discriminant DD₍, ₊ = disc₀䂵 discₓ f (x), which appears in the proof of the van der Waerden--Bhargava theorem. We conjecture that DD₍, ₊ is the product of a square, a cube, and possibly a linear monomial and we prove this when k=0. We also investigate the (typically large and smooth) outlying integer constant in the factorization of DD₍, ₊.

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Anderson et al. (Tue,) studied this question.

www.synapsesocial.com/papers/68d4739d31b076d99fa6bd5a — DOI: https://doi.org/10.48550/arxiv.2507.16138

Authors

Theresa C. Anderson

Adam Bertelli

Evan M. O’Dorney

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The structure of the double discriminant

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Abstract

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