Differential Equations for Algebraic Functions

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series

Integral zeros of a polynomial with linear recurrences as coefficients

Let $K$ be a number field, $S$ a finite set of places of $K$, and $\mathcal{O}_S$ be the ring of $S$-integers. Moreover, let $$G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)}$$ be a polynomial in $Z$ having simple linear recurrences of integers evaluated at $n$ as coefficients. Assuming some technical conditions we give a description of the zeros $(n,z) \in \mathbb{N} \times \mathcal{O}_S$ of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials.Comment: 13 page

Low Complexity Algorithms for Linear Recurrences

We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are hypergeometric over the rational numbers. The algorithms for these tasks all involve as an intermediate quantity an integer $N$ (dispersion or root of an indicial polynomial) that is potentially exponential in the bit size of their input. Previous algorithms have a bit complexity that is at least quadratic in $N$. We revisit them and propose variants that exploit the structure of solutions and avoid expanding polynomials of degree $N$. We give two algorithms: a probabilistic one that detects the existence or absence of nonzero polynomial and rational solutions in $O(\sqrt{N}\log^{2}N)$ bit operations; a deterministic one that computes a compact representation of the solution in $O(N\log^{3}N)$ bit operations. Similar speed-ups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.Comment: This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistributio