Stratifying Discriminant Hypersurface
Rizeng Chen, Hoon Hong, Jing Yang
TL;DR
This work addresses stratifying the discriminant hypersurface of a univariate polynomial by the number of distinct complex roots using discriminant data alone. It introduces two intrinsic approaches—the order-based stratification, which characterizes $Z_k$ as the zero locus of derivatives of $D$ up to order $n-k-1$, and the smoothness-based stratification, obtained by iteratively taking singular loci to align strata with fixed root multiplicities—complementing the classical subdiscriminant description. The main results connect root multiplicities to the order of vanishing of $D$ and to a hierarchy of singularities, providing intrinsic geometric descriptions and identifying irreducible components via coincident root loci. These findings deepen the understanding of discriminant geometry and its link to root structure, with potential extensions to real-root counting and multivariate systems.
Abstract
This paper investigates the stratification of the discriminant hypersurface associated with a univariate polynomial via the number of its distinct complex roots. We introduce two novel approaches different from the one based on subdiscriminants. The first approach stratifies the discriminant hypersurface by recursively removing all the lowest-order points, while the second one stratifies the discriminant hypersurface by recursively removing all the smooth points. Both approaches rely solely on the discriminant itself instead of using high-order subdiscriminants. These results offer new insights into the intrinsic geometry of the discriminant and its connection to root multiplicity.