On the strength of general polynomials

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 Fröberg'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 $9$.

19 pages, final revision

References in corpus (7)

• Progression-free sets in Z_4^n are exponentially small
• On cap sets and the group-theoretic approach to matrix multiplication
• Polynomials and tensors of bounded strength
• A note on Fröberg's conjecture for forms of equal degrees
• Topological noetherianity for cubic polynomials
• The set of forms with bounded strength is not closed
• Strength and Hartshorne's Conjecture in high degree

Cited by in corpus (2)

• Universality of high-strength tensors
• The set of forms with bounded strength is not closed