van Frankenhuijsen, Machiel ABC implies the radicalized Vojta height inequality for curves. (English) Zbl 1197.11087 J. Number Theory 127, No. 2, 292-300 (2007). Summary: The truncated or radicalized counting function of a meromorphic function \(f : \mathbb C \to \mathbb P ^{1}(\mathbb C)\) counts the number of times that \(f\) takes a value \(a\), but without multiplicity. By analogy, one also defines this function for numbers. In this sequel to the author’s paper in J. Number Theory 95, 289-302 (2002; Zbl 1083.11042), we prove the radicalized version of Vojta’s height inequality, using the ABC conjecture. We explain the connection with a conjecture of Serge Lang about the different error terms associated with Vojta’s height inequality and with the radicalized Vojta height inequality. Cited in 5 Documents MSC: 11J97 Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.) 11D45 Counting solutions of Diophantine equations 11D75 Diophantine inequalities 14G40 Arithmetic varieties and schemes; Arakelov theory; heights Keywords:ABC conjecture; the error term in the ABC conjecture; radicalized Vojta height inequality; Diophantine approximation; Roth’s theorem; type of an algebraic number; Mordell’s conjecture; effective Mordell Citations:Zbl 1083.11042 PDFBibTeX XMLCite \textit{M. van Frankenhuijsen}, J. Number Theory 127, No. 2, 292--300 (2007; Zbl 1197.11087) Full Text: DOI arXiv References: [1] Belyĭ, G. V., On Galois extensions of a maximal cyclotomic field, Math. USSR Izvestija, 14, 2, 247-256 (1980) · Zbl 0429.12004 [2] Huisman, J., Heights on abelian varieties, (Edixhoven, B.; Evertse, J. H., Diophantine Approximation and Abelian Varieties. Diophantine Approximation and Abelian Varieties, Lecture Notes in Math., vol. 1566 (1993), Springer-Verlag: Springer-Verlag New York), 51-61 · Zbl 0811.14026 [3] S. Lang, Questions about the error term of diophantine inequalities, Preprint, 2005; S. Lang, Questions about the error term of diophantine inequalities, Preprint, 2005 [4] Lang, S.; Cherry, W., Topics in Nevanlinna Theory, Lecture Notes in Math., vol. 1433 (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0709.30030 [5] Noguchi, J., On Nevanlinna’s second main theorem, (Noguchi, J.; etal., Geometric Complex Analysis (1996), World Scientific), 489-503 · Zbl 0927.32015 [6] Oesterlé, J., Nouvelles approches du “Théorème” de Fermat, Astérisque, 161-162, 165-186 (1988), Sém. Bourbaki 1987-1988 (694) [7] Roth, K. F., Rational approximation to algebraic numbers, Mathematika, 2, 1-20 (1955) · Zbl 0064.28501 [8] Stewart, C. L.; Tijdeman, R., On the Oesterlé-Masser conjecture, Mh. Math., 102, 251-257 (1986) · Zbl 0597.10042 [9] Stothers, W. W., Polynomial identities and hauptmoduln, Quart. J. Math. Oxford (2), 32, 349-370 (1981) · Zbl 0466.12011 [10] van Frankenhuijsen, M., The ABC conjecture implies Roth’s theorem and Mordell’s conjecture, Mat. Contemp., 16, 45-72 (1999) · Zbl 0977.11016 [11] van Frankenhuijsen, M., A lower bound in the ABC conjecture, J. Number Theory, 82, 91-95 (2000) · Zbl 0998.11033 [12] van Frankenhuijsen, M., The ABC conjecture implies Vojta’s height inequality for curves, J. Number Theory, 95, 289-302 (2002) · Zbl 1083.11042 [13] Vojta, P., Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math., vol. 1239 (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0609.14011 [14] Vojta, P., A more general ABC conjecture, Int. Math. Res. Not., 21, 1103-1116 (1998) · Zbl 0923.11059 [15] Vojta, P., Nevanlinna theory and Diophantine approximation, (Schneider, M.; Siu, Y.-T., Several Complex Variables. Several Complex Variables, Math. Sci. Res. Inst. Publ., vol. 37 (1999), Cambridge Univ. Press: Cambridge Univ. Press New York), 535-564 · Zbl 0960.32013 [16] P. Vojta, Letter to Elkies, 1992; P. Vojta, Letter to Elkies, 1992 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.