Computing solutions of linear Mahler equations

Abstract

Mahler equations relate evaluations of the same function f at iterated bth 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.

Desingularization in the q-Weyl algebra

“…Another direction of research we want to consider in the future is the desingularization problem for linear Mahler equations [23], which attracted quite some interest in the computer algebra community recently, see for example [8].…”

An algorithm to determine regular singular Mahler systems

“…In that case, the system is regular singular at 0 and K-equivalent to the identity matrix. From this point of view, Theorem 1 can be seen as a generalisation of the results of [CDDM18].…”

“…there exists an integer d ∈ D such that any solution y ∈ K of (11) belongs to Q z 1/d . Furthermore, as shown in [CDDM18], this integer d depends only on the valuation of the rational functions which are the coefficients of the Mahler equation (11). In particular, it does not depend on λ.…”

“…In [CDDM18], the authors built an algorithm to decide whether or not a Mahler system has a fundamental matrix of solutions in GL m (K). In that case, the system is regular singular at 0 and K-equivalent to the identity matrix.…”

“…Acknowledgement. -The authors would like to thank Thomas Dreyfus for his lights about the paper [CDDM18]. They also thank Boris Adamczewski and Julien Roques for the precious discussions about this work.…”