Computing solutions of linear Mahler equations

TL;DR

This paper introduces polynomial-time algorithms for solving linear Mahler equations involving series, polynomials, and rational functions, addressing degree blow-up issues and enabling applications in number theory and automatic sequences.

Contribution

The work presents the first polynomial-time algorithms for solving linear Mahler equations and computing the gcd of Mahler operators, improving computational efficiency.

Findings

Algorithms for solving Mahler equations in polynomial time

Effective methods for computing gcd of Mahler operators

Addresses exponential degree blow-up in Mahler operator applications

Abstract

Mahler equations relate evaluations of the same function $f$ at iterated $b$th powers of the variable. They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms. Recently, the problem of solving Mahler equations in closed form has occurred in connection with number-theoretic questions. A difficulty in the manipulation of Mahler equations is the exponential blow-up of degrees when applying a Mahler operator to a polynomial. In this work, we present algorithms for solving linear Mahler equations for series, polynomials, and rational functions, and get polynomial-time complexity under a mild assumption. Incidentally, we develop an algorithm for computing the gcrd of a family of linear Mahler operators.