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Mathematics of Computation

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Sums of heights of algebraic numbers

Author: Gregory P. Dresden
Journal: Math. Comp. 72 (2003), 1487-1499
MSC (2000): Primary 11R04, 11R06; Secondary 12D10
Published electronically: December 6, 2002
MathSciNet review: 1972748
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Abstract: For $A_t(x) = f(x) - t\, g(x)$, we consider the set $\{ \sum_{A_t(\alpha) = 0} h(\alpha) : t \in \overline{\mathbb{Q} } \}$. The polynomials $f(x), g(x)$ are in $\mathbb{Z} [x]$, with only mild restrictions, and $h(\alpha)$ is the Weil height of $\alpha$. We show that this set is dense in $[d, \infty)$ for some effectively computable limit point $d$.

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Additional Information

Gregory P. Dresden
Affiliation: Department of Mathematics, Washington & Lee University, Lexington, Virginia 24450-0303

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