The structure of the double discriminant
Theresa C. Anderson, Ufuoma V. Asarhasa, Adam Bertelli, Fabian Gundlach, Evan M. O'Dorney
TL;DR
The paper provides an explicit, purely algebraic factorization of the double discriminant $DD_{n,k}$ of the discriminant $D$ of a univariate polynomial $f(x)$. The main result expresses $DD_{n,k}$ as $DD_{n,k} = c_{n,k} R_0 R_n A_{n,k}^3 B_{n,k}^2$, with concrete definitions for $R_0$, $R_n$, $A_{n,k}$, and $B_{n,k}$, and a complete treatment of the cases $k=0$ and $1\le k\le n-1$. In the case $k=0$, $A_{n,0}=\mathrm{disc}(f')$ and $B_{n,0}$ is given by a product over the roots of $f'$, while for $1\le k\le n-1$ one works with $g(x)=x f'(x) - k f(x)$ and $F_0(x)=f(x)/x^k$ to define $A$ and $B$. The factors $A$ and $B$ encode root-multiplicity phenomena: $A$ vanishes at triple roots and $B$ vanishes when two double roots occur (or a quadruple root), and the paper proves a converse-type statement. The arithmetic of the constant term $c_{n,k}$ is analyzed, revealing structured divisibility by powers of two and small primes tied to $2\gcd(n,k)$, with conjectures and data supporting a broader pattern relevant to number theory and the Bhargava--van der Waerden theorem.
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.