zbMATH — the first resource for mathematics

Positive semigroups and generalized Frobenius numbers over totally real number fields. (English) Zbl 1448.11066
Summary: 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.
11D07 The Frobenius problem
11H06 Lattices and convex bodies (number-theoretic aspects)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11D45 Counting solutions of Diophantine equations
11G50 Heights
Full Text: DOI
[1] 10.1016/j.jnt.2006.05.020 · Zbl 1114.11025
[2] 10.1137/090778043 · Zbl 1211.90132
[3] 10.4064/aa155-1-5 · Zbl 1297.11014
[4] https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i2p42/0
[5] 10.1007/978-3-319-24298-9_29 · Zbl 1358.52016
[6] 10.1007/978-1-4419-9060-0_2
[7] 10.2307/2041711 · Zbl 0291.10014
[8] 10.1007/978-3-642-62035-5
[9] 10.1007/s00454-006-1295-2 · Zbl 1136.11307
[10] 10.1016/j.ejc.2010.11.001 · Zbl 1239.11024
[11] 10.1007/BF01204720 · Zbl 0753.11013
[12] 10.1093/acprof:oso/9780198568209.001.0001
[13] 10.1093/imrn/rnv268 · Zbl 1404.11082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.