# Counting bounded elements of a number field

## TL;DR

This paper provides estimates for the number and independence of elements in a number field with bounded valuations, and derives new bounds for ideal lattice minima using advanced mathematical techniques.

## Contribution

It introduces novel bounds for elements and lattice minima in number fields by combining group theory, ramification, and geometry of numbers.

## Key findings

- New bounds for the count of elements with valuation constraints
- Bounds on the maximal number of linearly independent elements
- Improved estimates for successive minima of ideal lattices

## 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.