Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

Abstract

The space Pol dCP d of all complex-valued binary forms of degree d (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition μ d. For each such stratum S μ , we introduce its secant degeneracy index μ which is the minimal number of projectively dependent pairwise distinct points on S μ , i.e., points whose projective span has dimension smaller than μ − 1. In what follows, we discuss the secant degeneracy index μ and the secant degeneracy index μ of the closureS μ .

Linearly dependent powers of binary quadratic forms

“…This problem has been studied in [8] and [14] without the degree condition on the summands. The recent [13] contains a generalization of this question, replacing f d i with j f a j ij for fixed tuples (a j ). If F is a W k (r, d) set, then there is an obvious way to transform the linear dependence of the d-th powers into a more natural expression for any m, 1 ≤ m ≤ r − 1: wheref j = (±λ j ) 1/d f j , for some p. In particular, a W k (2m, d) set addresses the classical question of parameterizing two equal sums of m d-th powers.…”