Simultaneous p-orderings and minimizing volumes in number fields

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• 作者：Jakub Byszewski^a ; ^{jakub.byszewski@uj.edu.pl} ; Mikołaj Fra̧czyk^b ; ^{mikolaj.fraczyk@math.u-psud.fr} ; Anna Szumowicz^a ; ^c ; ^{anna.szumowicz@imj-prg.fr}
• 关键词：Simultaneous orderings ; Integer-valued polynomials ; Generalized factorials ; Euler&ndash ; Kronecker constants ; Number fields
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：173
• 期：Complete
• 页码：478-511
• 全文大小：647 K
• 卷排序：173

文摘

In [VP], V.V. Volkov and F.V. Petrov consider the problem of existence of the so-called n  -universal sets (related to simultaneous pp-orderings of Bhargava) in the ring of Gaussian integers. A related problem concerning Newton sequences was considered by D. Adam and P.-J. Cahen in [AC]. 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.