On the rank of $n\times n$ matrix multiplication

TL;DR

This paper establishes new lower bounds on the rank of n×n matrix multiplication, improving previous bounds and providing tighter estimates for various values of p, with implications for computational complexity.

Contribution

The paper introduces improved lower bounds on matrix multiplication rank for all p ≤ n, surpassing earlier bounds and extending the range of n where these bounds are tighter.

Findings

New lower bound for matrix multiplication rank for all p ≤ n.

Improved bounds for n ≥ 132 and n ≥ 24 with specific p values.

Enhanced understanding of the complexity of matrix multiplication.

Abstract

For every $p\leq n$ positive integer we obtain the lower bound $(3-\frac{1}{p+1})n^2-\big(2\binom{2p}{p+1}-\binom{2p-2}{p-1}+2\big)n$ for the rank of the $n\times n$ matrix multiplication. This bound improves the previous one $(3-\frac{1}{p+1})n^2-\big(1+2p\binom{2p}{p}\big)n$ due to Landsberg. Furthermore our bound improves the classic bound $\frac{5}{2}n^2-3n$, due to Bl\"aser, for every $n\geq 132$. Finally, for $p = 2$, with a sligtly different strategy we menage to obtain the lower bound $\frac{8}{3}n^2-7n$ which improves Bl\"aser's bound for any $n\geq 24$.