We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers.

中文翻译：

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计算数字字段的有界元素

我们在数域中估计元素的数量和线性独立元素的最大数量，并对其估值有规定的界限。作为副产品，我们获得了理想格的连续最小值的新边界。我们的论点结合了群论、分枝论和数的几何。