Symmetric powers: structure, smoothability, and applications

TL;DR

This paper studies the structure and smoothability of symmetric powers of algebras, providing bounds on border ranks, proving smoothability, and exploring applications in complexity theory using the concept of encompassing polynomials.

Contribution

It introduces the notion of encompassing polynomials and proves smoothability of symmetric powers, along with optimal border rank bounds under mild conditions.

Findings

Symmetric powers of algebras are smoothable.

Upper bounds for border ranks are established and shown to be optimal.

Applications to complexity theory are demonstrated.

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. Many of the results rest on the notion of an encompassing polynomial, which we introduce.