Sums of heights of algebraic numbers

Author:
Gregory P. Dresden

Journal:
Math. Comp. **72** (2003), 1487-1499

MSC (2000):
Primary 11R04, 11R06; Secondary 12D10

DOI:
https://doi.org/10.1090/S0025-5718-02-01481-3

Published electronically:
December 6, 2002

MathSciNet review:
1972748

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For , we consider the set . The polynomials are in , with only mild restrictions, and is the Weil height of . We show that this set is dense in for some effectively computable limit point .

**1.**M.-J. Bertin,*The operator and the reciprocal integers*, Number theory (Ottawa, ON, 1996), Amer. Math. Soc., Providence, RI, 1999, pp. 17-23. MR**2000e:11131****2.**E. Bombieri, A. J. van der Poorten, and J. D. Vaaler,*Effective measures of irrationality for cubic extension of number fields*, Annali della Scuola Normale Superiore di Pisa Scienze Fisiche e Matematiche**23**(1996), no. 4, 211-248. MR**98d:11083****3.**Christophe Doche,*On the spectrum of the Zhang-Zagier height*, Math. Comp.**70**(2001), no. 233, 419-430. MR**2001f:11181****4.**-,*Zhang-Zagier heights of perturbed polynomials*, J. Théor. Nombres Bordeaux**13**(2001), no. 1, 103-110, 21st Journées Arithmétiques (Rome, 2001). MR**2002f:11080****5.**G. Dresden,*Spectra of heights over certain finite groups*, Ph.D. thesis, The University of Texas at Austin, August 1997.**6.**-,*Orbits of algebraic numbers with low heights*, Math. Comp.**67**(1998), no. 222, 815-820. MR**98h:11128****7.**V. Flammang,*Sur la longueur des entiers algébriques totalement positifs*, J. Number Theory**54**(1995), 60-72. MR**96i:11129****8.**V. Flammang,*Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers*, Math. Comp.**65**(1996), no. 213, 307-311. MR**96d:11124****9.**G. Rhin and C. J. Smyth,*On the Mahler measure of the composition of two polynomials*, Acta Arith.**79**(1997), no. 3, 239-247. MR**98b:11109****10.**A. Schinzel,*On the product of the conjugates outside the unit circle of an algebraic number*, Acta Arith.**24**(1973), 385-399. MR**50:12963**; MR**51:8070****11.**C. J. Smyth,*On the measure of totally real algebraic integers*, J. Austral. Math. Soc. (Series A)**30**(1980), 137-149. MR**82j:12002a****12.**-,*On the measure of totally real algebraic integers, II*, Math. Comp.**37**(1981), no. 155, 205-208. MR**82j:12002b****13.**G. Szegö,*Orthogonal polynomials*, American Mathematical Society, 1975. MR**51:8724****14.**D. Zagier,*Algebraic numbers close to both and*, Math. Comp.**61**(1993), no. 203, 485-491. MR**94c:11104****15.**S. Zhang,*Positive line bundles on arithmetic surfaces*, Ann. of Math. (2)**136**(1992), no. 3, 569-587. MR**93j:14024**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11R04,
11R06,
12D10

Retrieve articles in all journals with MSC (2000): 11R04, 11R06, 12D10

Additional Information

**Gregory P. Dresden**

Affiliation:
Department of Mathematics, Washington & Lee University, Lexington, Virginia 24450-0303

Email:
dresdeng@wlu.edu

DOI:
https://doi.org/10.1090/S0025-5718-02-01481-3

Received by editor(s):
May 24, 1999

Received by editor(s) in revised form:
December 10, 2001

Published electronically:
December 6, 2002

Additional Notes:
I would like to thank Dr. C. J. Smyth and Dr. J. Vaaler, and I would also like to thank the referee for helpful comments and an improved proof of Theorem 6.1.

Article copyright:
© Copyright 2002
American Mathematical Society