On the evaluation of some sparse polynomials

We give algorithms for the evaluation of sparse polynomials of the form P = p0 + p1x + p2x 4 + · · ·+ pN−1x 2 , for various choices of coefficients pi. First, we take pi = 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 √ 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 sub-quadratic time.