The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with $m$ matrix-matrix multiplications, but with an arbitrary number of matrix additions and scaling operations. We characterize this set through a tabular parameterization. By deriving equivalence transformations of the tabular representation, we establish new methods that can be used to construct elements of $\Pi_{2^{m}}^*$ and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by $m^2$, which is subsequently proven to be sharp, i.e., $\dim(\Pi_{2^m}^*)=m^2$. Consequently, we have identified a parameterization that, to the best of our knowledge, is the first minimal parameterization. We also conduct a study using computational tools from algebraic geometry to determine the largest degree $d$ such that all polynomials of that degree belong to $\Pi_{2^{m}}^*$, or its closure. In many cases, the computational setup is constructive in the sense that it can also be used to determine a specific evaluation scheme for a given polynomial.