An effective Roth's theorem for function fields
The famous Roth's theorem states that given an algebraic number \(\alpha\) and a positive \(\varepsilon\), for all but finitely many rationals \(p/q\) the inequality \[ |\alpha-p/q|>q^{-2-\varepsilon} \] holds. Works of Mahler and Ridout led to generalizations in three directions: (1) the field containing the approximations can be choosed to be any fixed number field; (2) the metrics involved can be any archimedean or \(p\)-adic metric; (3) the target \(\alpha\) can be replaced by a finite set of algebraic points. In the general case the term \(q^{-2-\varepsilon}\) is replaced by the same power of the Weil height of the approximant. The problem generalizes naturally to function fields: Take a smooth projective curve \({\mathcal C}\) over an algebraically closed field \(k\) of characteristic \(0\); with every point \(P\) of \({\mathcal C}\) associate the valuation \(v_P\) of the function field \(K=k({\mathcal C})\), which can be extended to the algebraic closure \(K^a\) of \(K\). The logarithmic height of an element \(f\in K\), denoted by \(h(f)\), is defined to be its degree (viewing \(f\) as a map to \({\mathbf P}_1\)). Roth's theorem for function fields states the following: Let \(S\) be a non empty finite subset of \({\mathcal C}\). For each \(P\in S\), let \(a_P\) be an element of \(K^a\). Suppose that \(\kappa\) is a real number \(>2\). Then the elements \(f\in K\) satisfying \[ \sum_{P\in S} \max\{0,v_P(f-a_P)\}\geq \kappa h(f) \] have bounded height. This theorem (in a slightly modified form) was proved by \textit{S. Uchiyama} [J. Fac. Sci., Hokkaido Univ., No. I 15, 173-192 (1961; Zbl 0098.03901)], following the pattern of Roth's original proof. The paper under review provides a remarkably simpler proof which makes use of Wronskian arguments. Moreover, the method of the proof enables one to obtain an explicit bound for the height of solutions to Roth's inequality. The number field analogue seems at present to be unreachable.
scientific article; zbMATH DE number 1026412