Schmidt, Wolfgang M. On Osgood’s effective Thue theorem for algebraic functions. (English) Zbl 0333.10019 Commun. Pure Appl. Math. 29, 759-773 (1976). Let \(K\) be the field of formal series \(\alpha = a_\rho t^\rho +a_{\rho-1} t^{\rho-1} + \cdots\), with \(\rho\) an integer and with coefficients in a field \(k\) of characteristic zero. Let \(\vert\alpha\vert\) be the non-archimedean absolute value with \(\vert\alpha\vert = e^\rho\) if \(\alpha_\rho \ne 0\). Recently, it was shown by C. F. Osgood [Nederl. Akad. Wet., Proc., Ser. A 78, 105–119 (1975; Zbl 0302.10034)] that if \(\alpha\) in \(K\) is algebraic of degree \(\nu\) over the rational function field \(k(t)\), and if \(\mu\) is an integer with \(\tfrac12 \nu < \mu < \nu\), then \[ \vert \alpha - p(t)/q(t) \vert \ge c\,\vert q(t)\vert^{-\mu}, \] for every rational function \(p(t)/q(t)\); here \(c=c(\alpha,\mu) >0\) is effectively computable. The author gives an explicit value of \(c\) in terms of \(\nu\), \(\mu\) and the height of \(\alpha\). Consider a “Thue equation” \(F(p,q)=m\) where \(m\ne 0\) is a polynomial in \(t\) and \(F\) is a form of degree \(> 2\) without multiple factors and coefficients which are polynomials in \(t\). The height \(\mathfrak H(F)\) is the maximum absolute value of the coefficients of \(F\). It is shown that, for every polynomial solution \(p = p(t)\), \(q = q(t)\) of the equation, \[ \vert p \vert, \vert q \vert \le \vert m \vert\, \mathfrak H(F)^{80}. \] Reviewer: Wolfgang M. Schmidt (Boulder, CO) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 5 Documents MSC: 11J61 Approximation in non-Archimedean valuations 11J68 Approximation to algebraic numbers 11D59 Thue-Mahler equations Keywords:Osgood’s effective Thue theorem; algebraic functions Citations:Zbl 0302.10034 PDFBibTeX XMLCite \textit{W. M. Schmidt}, Commun. Pure Appl. Math. 29, 759--773 (1976; Zbl 0333.10019) Full Text: DOI References: [1] Baker, Phil. Trans. Roy. Soc. Lon. Ser. A 263 pp 173– (1967/68) [2] Baker, Proc. Camb. Phil. Soc. 65 pp 439– (1969) [3] Kolchin, Proc. Am. Math. Soc. 10 pp 238– (1959) [4] Mahler, Ann. of Math. 42 pp 488– (1941) [5] Osgood, Mathematika 20 pp 4– (1973) [6] Osgood, Indag. Math. 37 pp 105– (1975) · doi:10.1016/1385-7258(75)90023-2 [7] Einfuhrung in die Transzendenten Zahlen Springer, Grundlehren, 81, 1957. · doi:10.1007/978-3-642-94694-3 [8] Thue, J. F. Math. 135 pp 284– (1909) [9] Uchiyama, J. Fac. Sci. Hokkaido Univ. Ser. 1 15 pp 173– (1961) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.