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2020 Positive semigroups and generalized Frobenius numbers over totally real number fields
Lenny Fukshansky, Yingqi Shi
Mosc. J. Comb. Number Theory 9(1): 29-41 (2020). DOI: 10.2140/moscow.2020.9.29

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.

Citation

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Lenny Fukshansky. Yingqi Shi. "Positive semigroups and generalized Frobenius numbers over totally real number fields." Mosc. J. Comb. Number Theory 9 (1) 29 - 41, 2020. https://doi.org/10.2140/moscow.2020.9.29

Information

Received: 30 July 2019; Revised: 3 November 2019; Accepted: 18 November 2019; Published: 2020
First available in Project Euclid: 20 March 2020

zbMATH: 07171953
MathSciNet: MR4066557
Digital Object Identifier: 10.2140/moscow.2020.9.29

Subjects:
Primary: 11D07, 11D45, 11G50, 11H06, 52C07

Rights: Copyright © 2020 Mathematical Sciences Publishers

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