Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the rank of $n\times n$ matrix multiplication   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 $n\geq 24$.

8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio