Linear recurrences with polynomial coefficients and computation of the Cartier-Manin operator on hyperelliptic curves

Abstract

We improve an algorithm originally due to Chudnovsky and Chudnovsky for computing one selected term in a linear recurrent sequence with polynomial coefficients. Using baby-steps/giant-steps techniques, the nth term in such a sequence can be computed in time proportional to √n, instead of n for a naive approach. As an intermediate result, we give a fast algorithm for computing the values taken by an univariate polynomial P on an arithmetic progression, taking as input the values of P on a translate on this progression. We apply these results to the computation of the Cartier-Manin operator of a hyperelliptic curve. If the base field has characteristic p, this enables us to reduce the complexity of this computation by a factor of order √p. We treat a practical example, where the base field is an extension of degree 3 of the prime field with p = 232 - 5 elements. © Springer-Verlag Berlin Heidelberg 2004.