The quest for non-commutative matrix multiplication algorithms in small dimensions has seen a lot of recent improvements recently. In particular, the number of scalar multiplications required to multiply two $4\times4$ matrices was first reduced in \cite{Fawzi:2022aa} from 49 (two recursion levels of Strassen's algorithm) to 47 but only in characteristic 2 or more recently to 48 in \cite{alphaevolve} but over complex numbers. We propose an algorithm in 48 multiplications with only rational coefficients, hence removing the complex number requirement. It was derived from the latter one, under the action of an isotropy which happen to project the algorithm on the field of rational numbers. We also produce a straight line program of this algorithm, reducing the leading constant in the complexity, as well as an alternative basis variant of it, leading to an algorithm running in $7 n^{2+\frac{\log_2 3}{2}} +o\left(n^{2+\frac{log_2 3}{2}}\right)$ operations over any ring containing an inverse of 2.

翻译：近年来，在小维度非交换矩阵乘法算法的研究方面取得了诸多进展。特别地，两个$4\times4$矩阵相乘所需的标量乘法次数首先在文献\\cite{Fawzi:2022aa}中从49次（Strassen算法的两层递归）降至47次，但仅限于特征2的情形；随后在文献\\cite{alphaevolve}中进一步降至48次，但要求运算在复数域上进行。本文提出一种仅需48次乘法且系数全为有理数的算法，从而消除了对复数的依赖。该算法由后者通过各向同性作用推导而得，该作用恰好将算法投影至有理数域上。我们还给出了该算法的直线式程序，降低了复杂度中的主导常数，并提供了其替代基变体，从而得到一个在任意包含2的逆元的环上运行复杂度为$7 n^{2+\frac{\log_2 3}{2}} +o\left(n^{2+\frac{log_2 3}{2}}\right)$次运算的算法。