Equations for lower bounds on border rank

TL;DR

This paper introduces new algebraic methods to find lower bounds on border rank of bilinear maps, applying them to matrix multiplication and confirming its border rank is at least seven.

Contribution

The paper develops novel techniques for identifying polynomials in the ideal of bilinear map varieties, providing new proofs for border rank bounds of matrix multiplication.

Findings

New polynomials do not vanish on M_2, confirming border rank is at least seven.

Methods applicable to various bilinear maps and their border ranks.

Provides a framework for implementing algebraic techniques in border rank analysis.

Abstract

We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r = 6 in the space of bilinear maps C^4 x C^4 -> C^4. This space of bilinear maps includes the matrix multiplication operator M_2 for two by two matrices. We show these newly obtained polynomials do not vanish on the matrix multiplication operator M_2, which gives a new proof that the border rank of the multiplication of 2 x 2 matrices is seven. Other examples are considered along with an explanation of how to implement the methods.