Computing solutions of linear Mahler equations

Frédéric Chyzak (1) , Thomas Dreyfus (2) , Philippe Dumas (1) , Marc Mezzarobba (3, 4)

1 SPECFUN - Symbolic Special Functions : Fast and Certified
2 IRMA - Institut de Recherche Mathématique Avancée
3 CNRS - Centre National de la Recherche Scientifique
4 PEQUAN - Performance et Qualité des Algorithmes Numériques

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.

Domaines: Calcul formel [cs.SC]

Frédéric Chyzak, Thomas Dreyfus, Philippe Dumas, Marc Mezzarobba. Computing solutions of linear Mahler equations. Mathematics of Computation, 2018, 87, pp.2977-3021. ⟨10.1090/mcom/3359⟩. ⟨hal-01418653v2⟩