Strength and partition rank under limits and field extensions

TL;DR

This paper studies how the complexity measures strength $s_K(f)$ and partition rank, along with their border variants, behave under field extensions and limiting processes for homogeneous polynomials and tensors.The authors introduce de-bordering results: for fixed degree $d$ and under $ ext{char}(K)=0$ or $>d$, $s_K(f)$ is bounded by a polynomial in the border strength $\ul{s}(f)$ (and similarly for partition rank), with explicit dependence on whether $K$ is infinite or finite.A key methodological advance is expressing low-border-rank forms through derivatives via the subspace $D(f)$, showing $f$ lies in a subalgebra generated by $O_d(r^d)$ (or $O_d(r^d\log r)$ over finite fields) elements of $D(f)$, and extending these ideas to collective and partition-rank variants using polynomial functor formalism and a Master theorem.The results unify and extend prior bounds on strength and partition rank, provide robust control over drops under extensions and jumps in limits, and connect to broader themes in higher-order Fourier analysis and algebraic geometry.

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.

Theorems & Definitions (41)

• Definition 1.0.1
• Example 1.1.1
• Theorem 1.1.2
• Definition 1.2.1
• Remark 1.2.2
• Theorem 1.2.3
• Theorem 1.3.1
• Definition 1.5.1
• Definition 1.5.2
• Theorem 1.5.3