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Schanuel's theorem for heights defined via extension fields

Abstract

Let kk be a number field, let θ\theta be a nonzero algebraic number, and let H()H(\cdot) be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of αk\alpha \in k with H(αθ)XH(\alpha \theta)\leq X. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of kk. This provides a conceptual framework for Loher and Masser's problem and generalizations thereof. Moreover, we analyze the leading constant in our asymptotic formula for Loher and Masser's problem. In particular, we prove a sharp upper bound in terms of the classical Schanuel constant.Comment: accepted for publication by Ann. Sc. Norm. Super. Pisa Cl. Sci., 201

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