Algebraic relations between values of Siegel \(E\)-functions and Mahler \(M\)-functions (Q6633611)

Let \(f_1,\dots,f_r\) be \(E\)-functions and \(\alpha\) a nonzero algebraic number. The authors prove that any homogeneous algebraic relation among the values \(f_1(\alpha),\dots,f_r(\alpha)\) either is \textit{mundane} (\textit{banale} in French), which means that it follows by specializing at \(\alpha\) a polynomial relation \(Q(z,f_1(z),\dots,f_r(z))=0\), homogeneous in \(f_1,\dots,f_r\) and with algebraic coefficients, satisfied by the functions, or else it arises by degenerating a homogeneous \textit{\(\delta\)-algebraic relation} among the functions \(f_1,\dots,f_r\), where \(\delta={\mathrm d}/{\mathrm d}z\). Here, the ideal of \(\delta\)-algebraic relations is the ideal of polynomials \(Q\) with coefficients in \(\overline{\mathbb{Q}}(z)\) in the variables \(X_{i,j}\), \(1\le i\le r\), \(j\ge 0\), such that the function obtained by substituting \(\delta^j(f_i)\) to the variable \(X_{i,j}\) is zero. Such a relation \textit{degenerates at \(\alpha\)} if \(Q(z,f_1(z),\dots,f_r(z))\) is not zero but the polynomial \(Q(\alpha,(X_{ij}))\) depends only on \(X_1,\dots,X_r\), where \(X_i=X_{i,0}\). The earlier results from Siegel-Shidlovskii and \textit{F. Beukers} [Ann. Math. (2) 163, No. 1, 369--379 (2006; Zbl 1133.11044)] had two extra assumptions which are relaxed here: the functions \(f_1,\dots,f_r\) were supposed to satisfy a linear differential system of equations, and the point \(\alpha\) was supposed to be a regular point for this system. \N\NThe authors prove a similar result for \(M_q\)-functions, which are power series with algebraic coefficients satisfying a \(\sigma_q\)-linear difference equation with coefficients in \(\overline{\mathbb{Q}}(z)\), where \(\sigma_q(f)(z)=f(z^q)\). For the proof of the result on \(M_q\)-functions the authors substitute to Beuker's refinement of the Siegel-Shidlovskii's Theorem the corresponding theorem due to \textit{P. Philippon} [J. Lond. Math. Soc., II. Ser. 92, No. 3, 596--614 (2015; Zbl 1391.11087)] and use a previous result of themselves [\textit{B. Adamczewski} and \textit{C. Faverjon}, Proc. Lond. Math. Soc. (3) 115, No. 1, 55--90 (2017; Zbl 1440.11132)].\N\NFrom their main result the authors deduce algorithms for determining all algebraic relations among the values of these functions. They also show that the relations which are not mundane are sporadic: the set of \(\alpha\) is finite. They deduce from their theorem some results due to \textit{S. Fischler} and \textit{T. Rivoal} [J. Éc. Polytech., Math. 11, 1--18 (2024; Zbl 1543.11061)] and obtain the \(M_q\)-analogs. They give examples of non-mundane relations involving Bessel functions for \(E\)-functions and the sequences of Thue-Morse, Baum-Sweet and Rudin-Shapiro for \(M_q\)-functions. \N\NThe point of view adopted by the authors reveals striking analogies between the theory of \(E\)-functions and that of \(M_q\)-functions.