Symmetric subrank and its border analogue

2026-04-14

AI Summary

This study investigates the asymptotic behavior and equality conditions of the symmetric subrank of homogeneous polynomials and its border analogue as the number of variables tends to infinity. By reducing polynomials to diagonal form via linear changes of variables and leveraging tools from geometric invariant theory, algebraic geometry, and tensor analysis, the work provides the first characterization of the asymptotic properties of these two ranks in high dimensions. The main contributions include establishing that for cubic (quartic) forms, the symmetric subrank coincides with its border counterpart whenever the latter is at most 3 (2), and systematically elucidating the asymptotic relationship between symmetric subrank and border symmetric subrank in the limit of infinitely many variables.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski for ordinary tensors, we determine the asymptotic behavior of symmetric subrank and symmetric border subrank of degree-d forms as the number of variables tends to infinity. Furthermore, by using results from geometric invariant theory we show that for cubic (resp. quartic) forms the symmetric subrank and symmetric border subrank coincide if the latter is at most three (resp. two).

Problem

• symmetric subrank
• border subrank
• homogeneous polynomial
• asymptotic behavior
• geometric invariant theory

Innovation

• symmetric subrank
• symmetric border subrank
• asymptotic behavior
• geometric invariant theory
• homogeneous polynomials