Hypertranscendency of first order Mahler functions (Q640884)

This paper presents a new application of difference Galois theory to hypertranscendency problems. Let \(R\) a ring equipped with a difference operator \(\sigma\) and a derivation \(\Delta\) with the same constant field \(k\) and such that \(\Delta \circ \sigma = p \sigma \circ \Delta\), where \(p \in k\).The author first gives a necessary condition for the equation \(\sigma y = a y + b\) to admit a hyperalgebraic solution. This criterion is then applied to the case of first order Mahler equations, where \(R\) is a ring of functions, \(\sigma\) sends \(f(x)\) to \(f(x^p)\) and \(\Delta\) sends \(f(x)\) to \(x f'(x)\). This in turn allows to recover a proof of a previous hypertranscendency result by \textit{K. Nishioka} [Aequationes Math. 27, 32--48 (1984; Zbl 0542.12012)].