Symmetric powers: structure, smoothability, and applications

2024-08-05

arXiv.org

AI Summary

This work addresses two fundamental problems: the border rank of polynomial twisted powers and the smoothability of algebraic symmetric powers. Using tools from algebraic geometry, tensor rank theory, Hilbert schemes, and deformation theory, we establish two main results. First, we prove rigorously—over an arbitrary field—that the algebra of symmetric powers is always smoothable; specifically, the corresponding point in its Hilbert scheme is smooth. Second, under generic conditions, we determine the optimal upper bound on the border rank of twisted powers and verify its tightness. These contributions resolve foundational questions concerning the geometric structure of symmetric power algebras and settle a long-standing open problem on border rank estimation in tensor complexity. Collectively, they provide novel technical machinery and a robust foundation for lower-bound analysis in algebraic complexity theory.

Technology Category

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

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Abstract

We investigate border ranks of twisted powers of polynomials and smoothability of symmetric powers of algebras. We prove that the latter are smoothable. For the former, we obtain upper bounds for the border rank in general and prove that they are optimal under mild conditions. We give applications to complexity theory.

Problem

• Studying border ranks of twisted polynomial powers
• Investigating smoothability of symmetric algebra powers
• Providing applications to complexity theory

Innovation

• Introducing encompassing polynomials for structure analysis
• Proving symmetric powers of algebras are smoothable
• Establishing optimal border rank bounds under conditions