Strength and slice rank of forms are generically equal

TL;DR

This paper proves that for generic homogeneous polynomials of degree at least 5 over an algebraically closed field of characteristic zero, the concepts of strength and slice rank are equivalent, extending previous results to higher degrees.

Contribution

It establishes the equality of strength and slice rank for generic polynomials of degree ≥ 5, and proves a conjecture on secant varieties of reducible polynomials.

Findings

Strength and slice rank are generically equal for degree ≥ 5.

Confirmed a conjecture on dimensions of secant varieties of reducible polynomials.

Extended known results from degrees 2-7 and 9 to higher degrees.

Abstract

We prove that strength and slice rank of homogeneous polynomials of degree $d \geq 5$ over an algebraically closed field of characteristic zero coincide generically. To show this, we establish a conjecture of Catalisano, Geramita, Gimigliano, Harbourne, Migliore, Nagel and Shin concerning dimensions of secant varieties of the varieties of reducible homogeneous polynomials. These statements were already known in degrees $2\leq d\leq 7$ and $d=9$.