On the rank of a symmetric form

TL;DR

This paper establishes a lower bound on the degree of finite apolar subschemes for symmetric forms, providing explicit formulas for monomials and linking minimal apolar length to tensor rank.

Contribution

It introduces a new lower bound for the degree of apolar subschemes based on the annihilator ideal, and characterizes minimal apolar length for monomials, connecting it to tensor rank.

Findings

Lower bound for apolar subscheme degree in terms of annihilator ideal generators

Explicit minimal apolar length formula for monomials

Minimal apolar length equals the rank when exponents are equal

Abstract

We give a lower bound for the degree of a finite apolar subscheme of a symmetric form F, in terms of the degrees of the generators of the annihilator ideal of F. In the special case, when F is a monomial x_0^d_0 x_2^d_2... x_n^d_n with d_0<= d_1<=...<=d_n-1<= d_n we deduce that the minimal length of an apolar subscheme of F is (d_0+1)...(d_n-1+1), and if d_0=..=d_n, then this minimal length coincides with the rank of F.