Simultaneous -orderings and minimising volumes in number fields
arXiv:1506.02696 · doi:10.1016/j.jnt.2016.09.010
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 -universal sets (related to simultaneous -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 -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.
new version, substantial corrections in section 6, will appear in Journal of Number Theory