Duality of Multiple Root Loci

TL;DR

This paper investigates the geometric properties of multiple root loci in univariate polynomials, focusing on their dual varieties, equations, parametrizations, and applications to Euclidean distance optimization and real rank analysis.

Contribution

It provides new insights into the equations, parametrizations, and ED degrees of dual varieties of multiple root loci, especially for hook partitions.

Findings

Computed ED degrees for hook partitions.

Identified dual hypersurfaces related to real rank boundaries.

Analyzed conormal varieties and projective duals of multiple root loci.

Abstract

The multiple root loci among univariate polynomials of degree $n$ are indexed by partitions of $n$. We study these loci and their conormal varieties. The projectively dual varieties are joins of such loci where the partitions are hooks. Our emphasis lies on equations and parametrizations that are useful for Euclidean distance optimization. We compute the ED degrees for hooks. Among the dual hypersurfaces are those that demarcate the set of binary forms whose real rank equals the generic complex rank.