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 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 $m^2$. Numerical simulations suggest that this is a sharp bound. Consequently, we have identified a parameterization, which, to our knowledge, is the first minimal parameterization. Furthermore, we 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.