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Geometric postulation of a smooth function and the number of rational points. (English) Zbl 0763.11025

This paper provides refinements and extensions of some of the results of E. Bombieri and the author [Duke Math. J. 59, 337-357 (1989; Zbl 0718.11048)]obtaining upper bounds for the number of integral points on the graphs of functions. Let \(f(x)\) be a sufficiently smooth function with graph \(\Gamma\). The author explores local conditions on \(f\) that control the multiplicity of the intersection of \(\Gamma\) with any algebraic curve of degree \(d\). He notes that his results apply to rational points on transcendental analytic functions defined on a closed bounded interval, so that, for example the number of rational points of height at most \(N\) on \(\Gamma\) is bounded by \(c(f,\epsilon)N^{\epsilon}\).
Of course there is much more. Suffice it here to say that the author cites results of considerable depth and great generality, yet obtained by techniques comprehensible to the unsophisticated reader.

MSC:

11G99 Arithmetic algebraic geometry (Diophantine geometry)
14G05 Rational points

Citations:

Zbl 0718.11048
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References:

[1] E. Bombieri and J. Pila, The number of integral points on arcs and ovals , Duke Math. J. 59 (1989), no. 2, 337-357. · Zbl 0718.11048 · doi:10.1215/S0012-7094-89-05915-2
[2] K. Mahler, Lectures on transcendental numbers , Springer-Verlag, Berlin, 1976. · Zbl 0332.10019 · doi:10.1007/BFb0081107
[3] 1 G. Pólya, On the mean-value theorem corresponding to a given linear homogeneous differential equation , Trans. Amer. Math. Soc. 24 (1922), no. 4, 312-324. JSTOR: · doi:10.2307/1988819
[4] 2 G. Pólya, Collected papers. Vol. III , Mathematicians of Our Time, vol. 21, MIT Press, Cambridge, MA, 1984. · Zbl 0561.01029
[5] G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II , Springer-Verlag, New York, 1976. · Zbl 0359.00003
[6] A. J. van der Poorten, Transcendental entire functions mapping every algebraic number field into itself , J. Austral. Math. Soc. 8 (1968), 192-193. · Zbl 0162.34803 · doi:10.1017/S144678870000522X
[7] W. M. Schmidt, Integer points on curves and surfaces , Monatsh. Math. 99 (1985), no. 1, 45-72. · Zbl 0551.10026 · doi:10.1007/BF01300739
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