Counting bounded elements of a number field
arXiv:1907.01116 · doi:10.1093/imrn/rnaa126
Abstract
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.
11 pages, LaTeX2e; v2: updated Corollary 3 and added Corollary 4; v3: revised version incorporating suggestions by the referees (the main changes are in the title and in the Introduction, e.g. Corollary 1 and the paragraph below Theorem 4 are new)