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arXiv · 2502.10007

Strength and partition rank under limits and field extensions

Arthur Bik·Jan Draisma·Amichai Lampert·Tamar Ziegler
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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.

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Arthur Bik, Jan Draisma, Amichai Lampert, Tamar Ziegler. 2025-02-14. Strength and partition rank under limits and field extensions. https://arxiv.org/abs/2502.10007

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