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.
We bound the tensor ranks of elementary symmetric polynomials, and we give explicit decompositions into powers of linear forms. The bound is attained when the degree is odd.
Finding the point in an algebraic variety that is closest to a given point is an optimization problem with many applications. We study the case when the variety is a Fermat hypersurface. Our formula for its Euclidean distance degree is a piecewise polynomial whose pieces are defined by subtle congruence conditons.2. ED-degree for Fermat hypersurfaces 2.1. Main theorem for Fermat hypersurfaces. In this section, we compute the ED-degree of F n,d for each n, d. Definition 2.1. For a positive integer p, fix a p-th primitive root of unity ζ. Define δ(m, p) to be the number of integer m-tuples (t 1 , . . . , t m ), 1 ≤ t i ≤ p, satisfying 1 + m i=1 ζ 2ti = 0.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.