Fast Coefficient Computation for Algebraic Power Series in Positive Characteristic

Abstract

We revisit Christol's theorem on algebraic power series in positive characteristic and propose yet another proof for it. This new proof combines several ingredients and advantages of existing proofs, which make it very well-suited for algorithmic purposes. We apply the construction used in the new proof to the design of a new efficient algorithm for computing the $N$th coefficient of a given algebraic power series over a perfect field of characteristic~$p$. It has several nice features: it is more general, more natural and more efficient than previous algorithms. Not only the arithmetic complexity of the new algorithm is linear in $\log N$ and quasi-linear in~$p$, but its dependency with respect to the degree of the input is much smaller than in the previously best algorithm. {Moreover, when the ground field is finite, the new approach yields an even faster algorithm, whose bit complexity is linear in $\log N$ and quasi-linear in~$\sqrt{p}$}.