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

For every $p\le n$ positive integer we obtain the lower bound $(3-\frac{1}{p+1})n^2-(2\left(\genfrac{}{}{0px}{}{2p}{p+1}\right)-\left(\genfrac{}{}{0px}{}{2p-2}{p-1}\right)+2)n$ for the rank of the $n\times n$ matrix multiplication. This bound improves the previous one $(3-\frac{1}{p+1})n^2-(1+2p\left(\genfrac{}{}{0px}{}{2p}{p}\right))n$ due to Landsberg. Furthermore our bound improves the classic bound $\frac{5}{2}{n}^2-3n$, due to Bl"aser, for every $n\ge 132$. 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 $n\ge 24$.