Improved lower bounds on the number of multiplications/divisions which are necessary to evaluate polynomials

Abstract

We improve some lower bounds which have been obtained by Strassen and Lipton. In particular there exist polynomials of degree n with 0–1 coefficients that cannot be evaluated with less than √n/(4 log n) nonscalar multiplications/divisions. The evaluation of (Formula Presented.) requires at least n/(12 log n) multiplications/divisions and at least √n/(8 log n) nonscalar multiplications/divisions. We specify polynomials with algebraic coefficients that require n/2 multiplications/divisions.