Uniformly counting points of bounded height

1. Introduction. In this paper we give some new uniform estimates for the cardinalities of certain sets involving algebraic numbers of bounded height. The estimates are nearly optimal with respect to the degree of the number eld. We mention some applications to problems about multiplicatively independent and dependent numbers in situations occurring in the recent theory of linear forms in logarithms associated with the name of Matveev. We also extend our counting estimates to algebraic vectors. Let K be a number eld of degree d = [K :Q] over the rational eld Q. We use the absolute height function dened for in K by

[1]  D. Bertrand,et al.  Duality on tori and multiplicative dependence relations , 1997, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[2]  D. Masser.  Counting points of small height on elliptic curves , 1989 .

[3]  W. Schmidt.  Northcott's theorem on heights II. The quadratic case , 1995 .

[4]  L. Kronecker,et al.  Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. , 1857 .

[5]  Thomas Ransford,et al.  Potential Theory in the Complex Plane: Bibliography , 1995 .

[6]  Jan-Hendrik Evertse,et al.  On equations inS-units and the Thue-Mahler equation , 1984 .

[7]  Wolfgang M. Schmidt,et al.  Diophantine Approximations and Diophantine Equations , 1991 .

[8]  S. Lang.  Algebraic Number Theory , 1971 .

[9]  D. Northcott.  An inequality in the theory of arithmetic on algebraic varieties , 1949, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  F. Amoroso,et al.  Le problème de Lehmer en dimension supérieure , 1999 .

[11]  E. Matveev,et al.  On Linear and Multiplicative Relations , 1994 .

[12]  A. Odlyzko.  Some analytic estimates of class numbers and discriminants , 1975 .

[13]  F. Thorne,et al.  Geometry of Numbers , 2017, Algebraic Number Theory.

[14]  E. Matveev,et al.  An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers , 1998 .

[15]  Stephen Hoel Schanuel.  Heights in number fields , 1979 .

[16]  G. Hardy,et al.  An Introduction to the Theory of Numbers , 1938 .

[17]  L. Tóth.  Lagerungen in der Ebene auf der Kugel und im Raum , 1953 .

[18]  C. A. Rogers.  The product ofn real homogeneous linear forms , 1950 .

[19]  H. Mulholland.  On the Product of n Complex Homogeneous Linear Forms , 1960 .

[20]  Stephanie Kleven.  Counting points of bounded height on del Pezzosurfaces , 2006 .

[21]  Michel Waldschmidt,et al.  Diophantine Approximation on Linear Algebraic Groups , 2000 .

[22]  Paul Voutier,et al.  An effective lower bound for the height of algebraic numbers , 2012, 1211.3110.

[23]  W. Schmidt.  Northcott's theorem on heights I. A general estimate , 1993 .

[24]  E. Matveev.  On the successive minima of the extended logarithmic height of algebraic numbers , 1999 .

[25]  H. F. Blichfeldt.  Note on the minimum value of the discriminant of an algebraic field , 1939 .

[26]  C. A. Rogers,et al.  An Introduction to the Geometry of Numbers , 1959 .