On the strength of general polynomials

TL;DR

This paper investigates the relationship between slice rank and strength of homogeneous polynomials, conjecturing they are equal for general forms and providing evidence up to degree 7 and 9.

Contribution

It proposes a conjecture relating slice rank and strength for general polynomials and verifies it for degrees up to 7 and 9 using a weakened form of Fröberg's Conjecture.

Findings

Slice rank bounds strength from above.

Conjecture holds for degrees up to 7 and 9.

Supports the conjecture with Hilbert series analysis.

Abstract

A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can be arbitrary large. However, in line with a conjecture by Catalisano et al. on the dimensions of secant varieties of the varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fr\"oberg's Conjecture on the Hilbert series of ideals generated by general forms, we show that our conjecture holds up to degree $7$ and in degree $9$.