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
