An essay on diophantine equations for analytic functions
If \(f[X_1,\dots, X_n]\) be a polynomial in \(X_1,\dots, X_n\) with coefficients in \(\mathbb{Z}\) or \(\mathbb{Z}[i]\). The question is if there is an algorithm which checks the existence of nonconstant analytic solutions \(x_1,\dots, x_n\) of the equation \(f=0\). The question can be asked in any prefixed ring of analytic functions, but important examples are the rings of functions analytic on the whole plane, on the closed and on the open unit disk. It is shown that the question has a positive answer (there is an algorithm) in the case of analytic functions on the disk -- open and closed. It is shown that the result follows from a ``transfer'' theorem of Ritt and the elimination theory of Seidenberg for algebraic differential equations. On the contrary, results of \textit{T. Pheidas} [J. Reine Angew. Math. 463, 153-167 (1995; Zbl 0819.03003)] prove that the question has a negative answer in the case of functions analytic on the plane. A similar negative result holds for functions on the disk, provided that one has a predicate for ``\(x\) is a constant function''. A number of interesting corollaries follow, e.g. the property ``\(x\) is a constant'', ``is not diophantine'' in the ring of functions analytic on the disk. The paper contains a discussion of various problems of complex analysis which relate to the question and gives a method for constructing nonconstant entire functions with prefixed properties of a diophantine nature, which works in some cases. We find it an opportunity to present two relevant and unpublished open questions of Rubel: Question 1: What about the above question where instead of analytic solutions one asks for solutions which are constructed from polynomials and the exponential function using the arithmetical operations and composition? Question 2: What about the question with multiplication replaced by composition, e.g. \(x\circ x+x+y\circ y=1\)?
existence of nonconstant analytic solutions
method for constructing nonconstant entire functions
scientific article; zbMATH DE number 783957