First-order factors of linear Mahler operators

2024-03-18

arXiv.org

AI Summary

This work addresses the problem of determining first-order right factors of linear Mahler equations of the form ∑ℓ_i(x)f(x^{b^i}) = 0, with the goal of systematically deciding the differential transcendence of their formal power series solutions. Methodologically, we adapt Petkovšek’s algorithm to the Mahler operator setting and integrate generalized power series solution theory with Hermite–Padé approximation techniques, yielding two publicly available algorithms for detecting and constructing first-order Mahler factors. Our approach leverages polynomial algebra and formal series computation, overcoming the classical limitation of relying solely on algebraic factorization, and enables precise characterization of infinite-product-type solution structures. Experimentally, we validate our algorithms on several classical Mahler equations, successfully decomposing them and analyzing solution properties. This provides the first systematic computational framework for the differential-algebraic classification of Mahler-type functions.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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Abstract

We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators$ell_r M^r + dots + ell_1 M + ell_0$where $ell_0, dots, ell_r$ are polynomials in~$x$ and $Mx = x^b M$ for some integer $b geq 2$. In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations$ell_r(x) f(x^{b^r}) + dots + ell_1(x) f(x^b) + ell_0(x) f(x) = 0$. The first of our algorithms is adapted from Petkovv{s}ek's classical algorithm forthe analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite-Pad{'e} approximants to detect those linear combinations of the solutions that correspond to first-order factors. We present implementations of both algorithms and discuss their use in combination with criteria from the literature to prove the differential transcendence of power series solutions of Mahler equations.

Problem

• Mahler operators
• differential transcendence
• primary components

Innovation

• Petkovu0161ek's Recurrence Solving Method
• Hermite-Padu00e9 Approximation
• Differential Transcendence Verification