Auxiliary functions in transcendence proofs

We discuss the role of auxiliary functions in the development of transcendental number theory. Earlier auxiliary functions were completely explicit (§ 1). The earliest transcendence proof is due to Liouville (§ 1.1), who produced the first explicit examples of transcendental numbers at a time where their existence was not yet known; in his proof, the auxiliary function is just a polynomial in one variable. Hermite’s proof of the transcendence of e (1873) is much more involved, the auxiliary function he builds (§ 1.2) is the first example of the Padé approximants (§ 1.3), which can be viewed as a far reaching generalization of continued fraction expansion [13, 14]. Hypergeometric functions (§ 1.4) are among the best candidates for using Padé approximations techniques. Another tool which plays the role of auxiliary functions is produced by interpolation formulae (§ 2). They occurred in the theory after a question by Weierstraß (§ 2.1) on the so-called exceptional set Sf of a transcendental function f , which is the set of algebraic numbers α such that f(α) is algebraic. The answer to his question is that any set of algebraic numbers is the exceptional set of some transcendental function f ; this shows that one should add further conditions in order to get transcendence criteria. One way is to replace algebraic number by rational integer: this gives rise to the study of integer–valued entire functions (§ 2.2) with the works of G. Pólya (1915), A.O. Gel’fond (1929) and many others. The connexion with transcendental number theory may not have been clear until the solution by A.O. Gel’fond in 1929 of the question of the transcendence of e, a special case of Hilbert’s seventh problem (§ 2.3). Along these lines, recent developments are due to T. Rivoal, who renewed forgotten rational interpolation formulae (1935) of R. Lagrange (§ 2.4). The simple (but powerful) construction by Liouville was extended to several variables by A. Thue (§ 3.1.1), who introduced the Dirichlet’s box principle (pigeonhole principle) (§ 3) into the topic of Diophantine approximation in the early 1900’s. In the 1920’s, Siegel (§ 3.1) developed this idea and applied it in 1932 to transcendental number theory. This gave rise to the Gel’fond–Schneider method (§ 3.1.2) which produces the Schneider–Lang Criterion in one (§ 3.1.3) or several (§ 3.1.4) variables. Among many developments of this method are results on modular functions (§ 3.1.6). Variants of the auxiliary functions produced by Dirichlet’s Box Principle are universal auxiliary functions, which have small Taylor coefficients at the origin (§ 3.2). Another approach, due to K. Mahler (§ 3.3), involves auxiliary functions whose existence is deduced from linear algebra instead of Thue–Siegel Lemma. In 1991 M. Laurent introduced interpolation determinants (§ 4). Two years