Positive semigroups and generalized Frobenius numbers over totally real number fields

Abstract

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.

Positive semigroups in lattices and totally real number fields

“…In this section, we let K be a totally real number field and use the notation of Section 1. Investigation of totally positive semigroups in totally real number fields from the standpoint of the generalized Frobenius problem, a somewhat different perspective, has been initiated in [3]. We can view an ideal I ⊆ O K as a lattice embedded into the Euclidean space R d via the Minkowski embedding…”