Strength and partition rank under limits and field extensions

TL;DR

This paper investigates how the strength and partition rank of polynomials and multilinear forms behave under field extensions and limits, establishing polynomial bounds relating these ranks.

Contribution

It proves that, under certain conditions, strength and partition rank are polynomially related to their border ranks, controlling their behavior under limits and field extensions.

Findings

Strength is at most polynomial in border strength for fixed degree.

Partition rank is similarly bounded in relation to border partition rank.

Results apply under mild characteristic conditions of the ground field.

Abstract

The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop under field extensions, and both can jump in a limit. We show that, for fixed degree and under mild conditions on the characteristic of the ground field, the strength is at most a polynomial in the border strength. We also establish an analogous result for partition rank. Our results control both the jump under limits and the drop under field extensions.