A Lower Bound for the Height of a Rational Function at S-unit Points

Abstract

Abstract. Let a, b be given multiplicatively independent positive integers and let ǫ > 0. In a recent paper written jointly also with Y. Bugeaud we proved the upper bound exp(ǫn) for gcd(a n − 1, b n − 1); shortly afterwards we generalized this to the estimate gcd(u − 1, v − 1) < max(|u|, |v|) ǫ , for multiplicatively independent S-units u, v ∈ Z. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u − 1, v − 1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G 2 m .

Rational periodic points for quadratic maps

“…We have dedicated an entire section to this case since it makes use of extra techniques. The main tools used in this section are some results proved by Corvaja and Zannier in [5] in which they obtained the estimate, for every fixed > 0, gcd(u − 1, v − 1) < max(H(u), H(v)) for all but finitely many multiplicatively independent S-units (where gcd denotes a suitable notion of greatest common divisor on number fields and H(·) denotes the multiplicative height). The results proved by Corvaja and Zannier in [5] also rest on the Subspace Theorem.…”

“…As already mentioned in the introduction, for the proof of Theorem 1.5 (the case n = 3), we need some results proved by P. Corvaja and U. Zannier in [5] which were obtained via the so called Subspace Theorem proved by W.M. Schmidt (e.g.…”

“…Below we state three of the results, proved by P. Corvaja and U. Zannier in [5], in a simpler form and adapted to our situation. 2 of the inequality …”

A cyclotomic generalization of the sequence \gcd (a^n-1,b^n-1)

“…It should be remarked that Corvaja and Zannier have made far-reaching generalizations of Theorem 1.1 in several directions, including function fields [5], [6], [7]. See also Luca [11].…”

Greatest common divisors of iterates of polynomials

“…The proof of Theorem 2 is quite different, as the tools used to prove Theorem 1 are no longer applicable to the case where both polynomials f and g are linear. The proof for this case relies heavily on diophantine methods, in particular an application of results from [CZ05], Roth's theorem, and a lemma of Siegel. These results are used to prove the case where everything is defined over Q, in Proposition 15.…”