Counting algebraic integers of fixed degree and bounded height
Let \(k\) be a number field and define, for positive integers \(n\) and \(e\), \[ k(n,e) = \{\boldsymbol{\beta}\in \overline{k}^n\, :\, [k(\boldsymbol{\beta}):k] = e\}. \] By a theorem of \textit{D. G. Northcott} [Proc. Camb. Philos. Soc. 45, 502--509, 510--518 (1949; Zbl 0035.30701)], any subset of \(k(n,e)\) of uniformly bounded height is finite. Let \(S\) be any subset of \(k(n,e)\) and \(H\) be the multiplicative Weil height. Asymptotic estimates for the counting function \[ N(S,{\mathcal H}) = |\{\boldsymbol{\beta}\in S\,:\, H(\boldsymbol{\beta}) \leq \mathcal H\}| \] have been found by Schanuel, Schmidt, Gao, Masser and Vaaler, Widmer for \(S=k(n,e)\) in many special cases. For \(S={\mathcal O}_k(n,e) := k(n,e)\cap{\mathcal O}_{\bar k}^n\), there are again results of Lang and Widmer for special cases and, in particular, there is a result of \textit{S. J. Chern} and \textit{J. D. Vaaler} [J. Reine Angew. Math. 540, 1--47 (2001; Zbl 0986.11017)] for the case \(S={\mathcal O}_{\mathbb Q}(1,e)\). The present paper is devoted to generalize the result of Chern and Vaaler [loc. cit.] to an arbitrary ground number field \(k\). Let \([k:{\mathbb Q}]=m=r+2s\), where \(r\) and \(2s\) are the number of real and complex embeddings of \(k\), respectively, and let \(q=r+s-1\). The main theorem states that \[ N({\mathcal O}_k(1,e),{\mathcal H}) = C_k^{(e)} {\mathcal H}^{me^2} (\log {\mathcal H})^{q-1} + \;\begin{cases} O({\mathcal H}^{me^2} (\log {\mathcal H})^{q-1}) & \text{if } q\geq 1, \\ O({\mathcal H}^{e(me-1)} {\mathcal L}) &\text{if } q=0, \end{cases} \] where \(C_k^{(e)}\) is an explicit constant and \({\mathcal L}= \log {\mathcal H}\) if \((m,e)=(1,2)\) and \({\mathcal L}=1\) otherwise.
scientific article; zbMATH DE number 6340806
scientific article