Hahn series and Mahler equations: algorithmic aspects (Q6586626)

Let \(\mathbb{K}\) be a field. A linear Mahler equation with coefficients in \(\mathbb{K}(z)\) is a functional equation \(a_{n}(z)y(z^{{\lambda}^{n}}) + a_{n-1}(z)y(z^{{\lambda}^{n-1}}) + \dots + a_{0}(z)y(z) = 0\) for some \({\lambda}\in\mathbb{Z}_{\geq 2}\), \(n\in\mathbb{Z}_{\geq 0}\), \(a_{i}(z)\in\mathbb{K}(z)\) (\(0\leq i\leq n\)), and \(a_{0}(z)a_{n}(z)\neq 0\). The authors give a positive answer to the question whether there is an algorithm to calculate the Hahn series solutions of a given linear Mahler equation. The field \(\mathcal{H}\) of Hahn series with coefficients in the field \(\mathbb{K}\) and with value group \(\mathbb{Q}\) consists of elements \((f_{\gamma})_{\gamma\in\mathbb{Q}}\) whose support \(\operatorname{supp}(f_{\gamma})_{\gamma\in\mathbb{Q}}\) is well ordered, that is, such that any nonempty subset of this support has a least element. An element \((f_{\gamma})_{\gamma\in\mathbb{Q}}\in\mathcal{H}\) is denoted by \(f = \sum_{\gamma\in\mathbb{Q}}f_{\gamma}z^{\gamma}\); the addition is defined in a natural way and if \(g = \sum_{\gamma\in\mathbb{Q}}g_{\gamma}z^{\gamma}\in\mathcal{H}\), then \(fg = \sum_{\gamma\in\mathbb{Q}}\left(\sum_{\gamma'+\gamma'' = \gamma}f_{\gamma'}g_{\gamma''}\right)z^{\gamma}\).\N\N The main result of the paper is an algorithm for computing Hahn series solutions of linear Mahler equations. The authors illustrate the steps of the algorithm by several examples, evaluate the complexity and demonstrate the work of the algorithm by applying it to the Rudin-Shapiro Mahler equation.