Simultaneous $p$-orderings and minimising volumes in number fields

TL;DR

This paper extends the concept of simultaneous p-orderings to all imaginary quadratic number fields, providing a counterexample to a conjecture on minimal universal sets and linking the problem to Euler-Kronecker constants.

Contribution

It generalizes previous results from Gaussian integers to all imaginary quadratic fields and disproves a conjecture on minimal universal set sizes.

Findings

Existence of strong counterexamples to the conjecture.

Established a lower bound on Euler-Kronecker constants.

Linked universal sets with Euler-Kronecker constants.

Abstract

In the paper "On the interpolation of integer-valued polynomials" (Journal of Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the problem of existence of the so-called $n$-universal sets (related to simultaneous $p$-orderings of Bhargava) in the ring of Gaussian integers. We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov-Petrov on minimal cardinality of $n$-universal sets. Along the way, we discover a link with Euler-Kronecker constants and prove a lower bound on Euler-Kronecker constants which is of the same order of magnitude as the one obtained by Ihara.