Schanuel's theorem for heights defined via extension fields

作者：Frei Christopher^*; Widmer Martin^*

来源：Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2016, 15: 355-398.

Let k be a number field, let theta be a nonzero algebraic number, and let H(.) 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 alpha is an element of k with H(alpha 0) <= X, and we analyze the leading constant in our asymptotic formula. In particular, we prove a sharp upper bound in terms of the classical Schanuel constant.
We also prove an asymptotic counting result for a new class of height functions defined via extension fields of k with a fairly explicit error term. This provides a conceptual framework for Loher and Masser's problem and generalizations thereof.
Finally, we establish asymptotic counting results for varying theta, namely, for the number of root p alpha of bounded height, where alpha is an element of k and p is any rational prime inert in k.

• 出版日期2016