On the Evaluation of Powers and Monomials

Let Yl, , Yp be monomials over the indeterminates Xl, , xq. For every y (yl, , yp) there is some minimum number L(y) of multiplications sufficient to compute y 1, , Yo from x 1, , xq and the identity 1. Let L(p, q, N) denote the maximum of L(y) over all y for which the exponent of any indeterminate in any monomial is at most N. We show that if p (N+ 1) () and q (N+ 1)(), then L(p, q, N) min {p, q} log N + H/log H + o(H/log H), where H pq log (N + 1) and all logarithms have base 2.