Strength and partition rank under limits and field extensions

2025-02-14

AI Summary

This paper investigates the stability of strength and partition rank for multivariate homogeneous polynomials and multilinear forms under field extensions and limiting processes. For fixed degree and arbitrary field characteristic—including non-algebraically closed fields—it establishes, for the first time, a polynomial upper bound on strength in terms of border strength, thereby uniformly controlling both rank jumps in limits and rank drops under field extension. The approach integrates tools from algebraic geometry, tensor rank theory, border rank analysis, and explicit field extension constructions. Key contributions are: (1) an explicit polynomial upper bound expressing strength as a function of border strength; (2) a complete resolution of the quantitative stability problem for strength and partition rank over non-algebraically closed fields; and (3) the first unified framework for asymptotic analysis of high-dimensional polynomial structures via controlled rank behavior.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Control strength drop in field extensions
• Manage rank jump under limits
• Establish polynomial bounds for ranks

Innovation

• Polynomial strength bounded by border strength
• Partition rank controlled under field extensions
• Limits and extensions analyzed for rank stability