The set of forms with bounded strength is not closed

TL;DR

This paper demonstrates that the set of homogeneous polynomials with bounded strength is not necessarily closed in the Zariski topology, especially for quartic forms with strength at most 3 in many variables.

Contribution

It proves that the set of forms with bounded strength is not always Zariski-closed, using polynomial functors, for certain quartic forms over algebraically closed fields.

Findings

The set of quartic forms with strength ≤ 3 is not Zariski-closed in many variables.

Polynomial functors are used to establish non-closure results.

The non-closure property depends on the number of variables and the field.

Abstract

The strength of a homogeneous polynomial (or form) is the smallest length of an additive decomposition expressing it whose summands are reducible forms. Using polynomial functors, we show that the set of forms with bounded strength is not always Zariski-closed. More specifically, if the ground field is algebraically closed, we prove that the set of quartics with strength $\leq3$ is not Zariski-closed for a large number of variables.