On the local dimensions of solutions of Brent equations

Let $\langle m,n,p \rangle$ be the matrix multiplication tensor. The solution set of Brent equations corresponds to the tensor decompositions of $\langle m,n,p \rangle$. We study the local dimensions of solutions of the Brent equations over the field of complex numbers. The rank of Jacobian matrix of Brent equations provides an upper bound of the local dimension, which is well-known. We calculate the ranks for some typical known solutions, which are provided in the databases \cite{Faw22+} and \cite{Heule19}. We show that the automorphism group of the natural algorithm computing $\langle m,n,p \rangle$ is $(\mathcal{P}_m\times \mathcal{P}_n\times \mathcal{P}_p)\rtimes Q(m,n,p)$, where $\mathcal{P}_m$, $\mathcal{P}_n$ and $\mathcal{P}_p$ are groups of generalised permutation matrices, $Q(m,n,p)$ is a subgroup of $S_3$ depending on $m$, $n$ and $p$. For other algorithms computing $\langle m,n,p \rangle$, some conditions are given, which imply the corresponding automorphism groups are isomorphic to subgroups of $(\mathcal{P}_m\times \mathcal{P}_n\times \mathcal{P}_p)\rtimes Q(m,n,p)$. So under these conditions, $m^2+n^2+p^2-m-n-p-3$ is a lower bound for the local dimensions of solutions of Brent equations. Moreover, the gap between the lower and upper bounds is discussed.