Linear Recurrences with Polynomial Coefficients and Computation of the Cartier-Manin Operator on Hyperelliptic Curves

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 = 2 − 5 elements.