Heights of algebraic points lying

on curves or hypersurfaces

Author:
Wolfgang M. Schmidt

Journal:
Proc. Amer. Math. Soc. **124** (1996), 3003-3013

MSC (1991):
Primary 11G30

DOI:
https://doi.org/10.1090/S0002-9939-96-03519-8

MathSciNet review:
1343724

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Abstract | References | Similar Articles | Additional Information

Abstract: Our first aim will be to give an explicit version of a generalization of the results of Zhang and Zagier on algebraic points with . Secondly, we will show that distinct algebraic points lying on a given curve of certain type can be distinguished in terms of some height functions. Thirdly, we will derive a bound for the number of points on such a curve whose heights are under a given bound and whose coordinates lie in a multiplicative group of given rank.

**1.**E. Dobrowolski,*On a question of Lehmer and the number of irreducible factors of a polynomial*, Acta Arith.**34**(1979), 391--401. MR**80i:10040****2.**D. H. Lehmer,*Factorization of certain cyclotomic functions*, Ann. Math.**34**(2) (1933), 461--479.**3.**H. P. Schlickewei,*Equations*, Annals of Math., (to appear).**4.**H. P. Schlickewei and W. M. Schmidt,*Linear equations in variables which lie in a multiplicative group*, In preparation.**5.**H. P. Schlickewei and E. Wirsing,*Lower bounds for the heights of solutions of linear equations*, Invent. Math, (to appear).**6.**W. M. Schmidt,*Diophantine Approximation*, Springer Lecture Notes in Mathematics**785**(1980). MR**81j:10038****7.**D. Zagier,*Algebraic numbers close to both 0 and 1*, Math. Computation**61**(1993), 485--491. MR**94c:11104****8.**S. Zhang,*Positive line bundles on arithmetic surfaces*, Ann. of Math.**136**(1992), 569--587. MR**93j:14024**

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Additional Information

**Wolfgang M. Schmidt**

Affiliation:
Department of Mathematics, University of Colorado, Boulder, Colorado 80309-0395

Email:
Schmidt@Euclid.colorado.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03519-8

Received by editor(s):
March 27, 1995

Additional Notes:
The author was supported in part by NSF grant DMS–9401426.

Communicated by:
William W. Adams

Article copyright:
© Copyright 1996
American Mathematical Society