# Polynomials and the exponent of matrix multiplication

## TL;DR

This paper introduces polynomial-based tensors, including the symmetrized matrix multiplication tensor, to leverage algebraic geometry techniques for studying the matrix multiplication exponent , aiming to deepen understanding of computational complexity.

## Contribution

It defines polynomial tensors with the same exponent as matrix multiplication and explores algebraic geometry methods to analyze , especially through the symmetrized tensor sM_n.

## Key findings

- Polynomials can represent matrix multiplication tensors with the same exponent .
- Algebraic geometry techniques can be applied to study  using these polynomial tensors.
- The symmetrized tensor sM_n provides a new perspective on matrix multiplication complexity.

## Abstract

We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix $A$ by $sM_n(A)=trace(A^3)$. The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent $\omega$.