Nondefective secant varieties of varieties of completely decomposable forms

TL;DR

This paper investigates the dimensions of secant varieties of completely decomposable forms, providing new cases where these varieties are nondefective, thus advancing understanding of a classical problem related to polynomial decompositions.

Contribution

The authors prove several new cases of nondefectiveness for secant varieties of completely decomposable forms, extending known results and proposing conjectures for broader classes.

Findings

Secant varieties are nondefective when s ≤ s_1(d) for n ≥ 3.

Secant varieties are nondefective when s ≥ s_2(d) for n=3.

Secant varieties are nondefective when s ≤ 2^{n-3}c(n,d) for d ≥ n ≥ 4.

Abstract

A variation of Waring's problem from classical number theory is the question, ``What is the smallest number $s$ such that any generic homogeneous polynomial of degree $d$ in $n+1$ variables may be written as the sum of at most $s$ products of linear forms?'' This question may be answered geometrically by determining the smallest $s$ such that the $s$\nth secant variety of the variety of completely decomposable forms fills the ambient space. If this secant variety has the expected dimension, it is called nondefective, and $s=\left\lceil\binom{n+d}{d}/(dn+1)\right\rceil$. It is conjectured that the secant variety is always nondefective unless $d=2$ and $2\leq s\leq\frac{n}{2}$. We prove several special cases of this conjecture. In particular, we define functions $s_1$ and $s_2$ such that the secant variety is nondefective when $n\geq 3$ and $s\leq s_1(d)$ or when $n=3$ and $s\geq s_2(d)$ and a function $c$ such that the secant variety is nondefective when $d\geq n\geq 4$ and $s\leq 2^{n-3}c(n,d)$. We further show that the secant variety is nondefective when $s\leq 30$ unless $d=2$ and $2\leq s\leq\frac{n}{2}$.