Let \(K\) be a field and \(F\) a field extension of \(K\) endowed with an endomorphism \(\phi\) such that \(\phi(K)\subset K\). A linear \(\phi\)-difference equation over \(K\) is an equation of the form \(\phi^{n}(y) + a_{n-1}\phi^{n-1}(y)+\dots+a_{0}y = 0\) where \(a_{0},\dots, a_{n-1}\in K\). The paper under review is concerned with the situation when one has a tower of field extensions \(\mathbb{C}\subset K\subset F\) where \(K\) is equipped with a pair of automorphisms \((\phi, \sigma)\) that extend over \(F\). The authors consider the following three cases.
(2S): \(K = \mathbb{C}(x)\), \(F = \mathbb{C}((x^{-1}))\) (the field of Laurent series) with shift operators \(\phi(x) = x+h_{1}\), \(\sigma(x) = x+h_{2}\) where \(h_{1},h_{2}\in\mathbb{C} \) and \(h_{1}/h_{2}\notin\mathbb{Q}\).
(2Q): \(K = \cup_{j\geq 1}\mathbb{C}(x^{1/j})\) (the field of ramified rational functions also denoted by \(\mathbb{C}(x^{1/\ast})\)), \(F = \cup_{j\geq 1}\mathbb{C}((x^{1/j}))\) (the field of Puiseux series also denoted by \(\mathbb{C}((x^{1/\ast}))\)) and the pair of automorphisms \((\phi, \sigma)\) is defined by \(\phi(x) = q_{1}\) and \(\sigma(x) = q_{2}x\), where \(q_{1}, q_{2}\in\mathbb{C}\) and whenever \(q_{1}^{n_{1}}q_{2}^{n_{2}} = 1\) (\(n_{1}, n_{2}\in\mathbb{Z}\)), one has \(n_{1}=n_{2}=0\) (then \(q_{1}\) and \(q_{2}\) are said to be multiplicatively independent). Another restriction in this case is that \(q_{1}\) and \(q_{2}\) cannot both be algebraic numbers of modulus \(1\) whose Galois conjugates all have modulus \(1\).
(2M): \(K=\mathbb{C}(x^{1/\ast})\), \(F = \mathbb{C}((x^{1/\ast}))\) and the pair of automorphisms \((\phi, \sigma)\) is defined by \(\phi(x) = x^{p_{1}}\), \(\sigma(x) = x^{p_{2}}\) where \(p_{1}\) and \(p_{2}\) are two multiplicatively independent natural numbers.
As it is proved in [R. Schäfke and M. F. Singer, J. Eur. Math. Soc. (JEMS) 21, No. 9, 2751–2792 (2019; Zbl 1425.39003)], if \(K\), \(F\) and \((\phi, \sigma)\) are defined as in cases (2S), (2Q), and (2M), then an element \(f\in F\) cannot satisfy both a linear \(\phi\)-difference equation and a linear \(\sigma\)-difference equation with coefficients in \(K\) unless \(f\in K\). The main result of the paper under review gives an essential generalization of this theorem, as well as of several statement of [T. Dreyfus et al., Math. Z. 298, No. 3–4, 1751–1791 (2021; Zbl 1476.39007)]: it is proven that if \(K\), \(F\) and \((\phi, \sigma)\) are defined as in cases (2S), (2Q), (2M) and \(f\in F\) is a solution to a linear \(\phi\)-difference equation over \(K\), then either \(f\in K\) or \(f\) is \(\sigma\)-transcendental over \(K\).
As a consequence of this theorem, the authors obtain the following result. Let \(K\), \(F\) and \((\phi, \sigma)\) be defined as in cases (2S), (2Q), (2M), let \(f\in F\) be a solution to a linear \(\phi\)-difference equation over \(K\) and let \(g\in F\) be a solution of a \(\sigma\)-difference equation over \(K\). Then \(f\) and \(g\) are algebraically independent over \(K\) unless \(f\in K\) or \(g\) is algebraic over \(K\). As consequences of the obtained results, the authors settled a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987 and gave an application to the algebraic independence of \(q\)-hypergeometric functions.