On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials

We present algorithms which use only O(x/ nonscalar multiplications (i.e. multiplications involving "x" on both sides) to evaluate polynomials of degree n, and proofs that at least x/ are required. These results have practical application in the evaluation of matrix polynomials with scalar coefficients, since the "matrix matrix" multiplications are relatively expensive, and also in determining how many multiplications are needed for polynomials with rational coefficients, since multiplications by integers can in principle be replaced by several additions.