Vol. 9, No. 1, 2020

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Positive semigroups and generalized Frobenius numbers over totally real number fields

Lenny Fukshansky and Yingqi Shi

Vol. 9 (2020), No. 1, 29–41
Abstract

The Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer’s conjecture for algebraic vectors satisfying some special conditions. We also use a result of Borosh and Treybig to obtain bounds on the size of representations and number of elements of bounded height in such positive semigroups of totally real algebraic integers.

Keywords
linear Diophantine problem of Frobenius, lattice points in polyhedra, affine semigroups, totally real number fields, heights
Mathematical Subject Classification 2010
Primary: 11D07, 11H06, 52C07, 11D45, 11G50
Milestones
Received: 30 July 2019
Revised: 3 November 2019
Accepted: 18 November 2019
Published: 8 January 2020
Authors
Lenny Fukshansky
Department of Mathematical Sciences
Claremont McKenna College
Claremont, CA
United States
Yingqi Shi
Department of Mathematical Sciences
Claremont McKenna College
Claremont, CA
United States