The core of this note is a study of those transformations \[ y\quad \mapsto \quad U(y)(x)=\sum^{s}_{i=1}R_ i(x)y(\frac{a_ ix}{1+b_ ix}) \] with the \(R_ i\) rational functions, for which the rationality of the function U (say, that it be an element of \({\mathbb{Q}}(x))\) and some suitable a priori restriction on y (say, that it have integer coefficients) entails the rationality of the series y. The author shows, in effect, that it suffices for the complex constants \(a_ i\) and \(b_ i\) to be such that the intersection of the subsets of \({\mathbb{C}}\) containing the origin and stable under all the maps \(x\mapsto a_ ix+b_ i\) has transfinite diameter strictly smaller than 1; a special case is if for all i one has \(| a_ i| +| b_ i| <1\).
Though somewhat disguised, the underlying problem here is a generalization of the matter of the Hadamard quotient of rational functions: if both \(\sum c_ hX^ h\) and \(\sum b_ hX^ h\) represent rational functions and the series \(\sum (c_ h/b_ h)X^ h\) could ‘possibly’ represent a rational function (for example, if all the quotients \(a_ h=(c_ h/b_ h)\) are rational integers) then \(\sum a_ hX^ h\) does indeed represent a rational function. This matter is now completely settled in the affirmative; see the reviewer’s paper [C.R. Acad. Sci., Paris, Sér. I 306, No.3, 97-102 (1988; Zbl 0635.10007)]. It would be interesting to understand this result in a wider setting. An appropriate viewpoint is hinted at in the present paper; the point of difficulty remaining being the boundary situation when the transfinite diameter is 1. It may be that R. S. Rumely’s very general “Capacity theory on algebraic curves” (to appear in the Springer Lecture Notes) has a rôle to play in this context.