The structure of the double discriminant

Abstract

For a polynomial $f(x)={\sum }_{i=0}^n{a}_i{x}^i$, we study the double discriminant $D{D}_{n,k}={\mathrm{disc}}_{a_k}{\mathrm{disc}}_{x}f(x)$. 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: $D{D}_{n,k}$ 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.

13 pages. Amplified with proofs of results conjectured in previous version