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The Thue-Siegel-Roth theorem for values of algebraic functions

0. The Thue-Siegel-Roth theorem [1], [2] on the approximation of algebraic numbers by rational numbers states that for a given algebraic number a and any 0 there exist only finitely many rational approximations p/q to a such that la-p/qllql--. Here, as well as in earlier versions. [3], the results are ineffective; they depend on the knowledge of at least one good approximation to a. Such good approximations are known only for an a of a special form, see [4] and [5]. Here we prove an entirely new theorem on appro.ximations of algebraic numbers, by considering diophantine approximations to values of algebraic functions. Our result shows that for an algebraic functions defined over Q(x) and regular at x=0, the "Roth" theorem holds for the number f(r) with a rational r:/:0 close to 0. Our methods are based on the Wronskian technique developed in [6] for the functional version of Roth’s theorem. A complete proof is presented for cubic algebraic functions, satisfying Ricatti equations. 1. Let f(x) be an algebraic function over Q(x) defined as. a solution of an algebraic equation P(x, f(x))=0 for an absolutely irreducible polynomial P(x, y) over Q[x, y]. We also assume that f(x) is regula.r at x=0 and has the Taylor expansion f(x)= --0 ax with aneQ. Theorem 1. Let f(x) be as above and let r a/ b for rational integers a and b. For every 0 there exist effective constants c --c(, f) 0 and c.-c. (, a, b, f) with the following properties. If b ]’_ c. a (+’), then