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.

翻译：我们研究如何以尽可能少的矩阵-矩阵乘法计算矩阵多项式 $p(X)$ 的问题，其中 $X$ 为大矩阵。更精确地说，令 $ \Pi_{2^{m}}^* $ 表示可通过 $m$ 次矩阵-矩阵乘法（但允许任意次矩阵加法和标量乘法）计算的多项式集合。我们通过表格参数化刻画该集合。通过推导表格表示的等价变换，我们建立了可用于构造 $ \Pi_{2^{m}}^* $ 元素并确定该集合一般性质的新方法。这些变换使我们能够消去变量，并证明其维度以 $m^2$ 为界。数值模拟表明该界是紧的。因此，我们得到了一种参数化方法，据我们所知，这是首个最小参数化表示。此外，我们利用代数几何的计算工具进行研究，以确定最大次数 $d$，使得所有该次数的多项式都属于 $ \Pi_{2^{m}}^* $ 或其闭包。在许多情况下，该计算框架具有构造性，即它也可用于为给定多项式确定具体的求值方案。