On the evaluation of some sparse polynomials

Abstract

We give algorithms for the evaluation of sparse polynomials of the form P=p0 + p1 x + p2 x^4 + ... + p_{n-1} x^{(N-1)^2} for various choices of coefficients . First, we take p_i=p^i, for some fixed p; in this case, we address the question of fast evaluation at a given point in the base ring, and we obtain a cost quasi-linear in sqrt{N}. We present experimental results that show the good behavior of this algorithm in a floating-point context, for the computation of Jacobi theta functions.

Next, we consider the case of arbitrary coefficients; for this problem, we study the question of multiple evaluation: we show that one can evaluate such a polynomial at N values in the base ring in subquadratic time.