Nonhomogeneous patterns on numerical semigroups. (Q2854968)

Numerical semigroups \(\Lambda\) are submonoids of the natural numbers with only finitely many missing numbers. The size of the gap is called the genus and the first nonzero element the multiplicity. The largest number not in the semigroup is called the Frobenius number.NEWLINENEWLINE The authors of this paper define the notion of a pattern by taking a polynomial \(P\) in \(n\) variables with integer coefficients and saying that a numerical semigroup admits the pattern if for any elements \(x_1,\ldots,x_n\in\Lambda\) with \(x_1\geq\cdots\geq x_n\), \(P(x_1,\ldots,x_n)\) is again in \(\Lambda\). This is a generalisation of the so called Arf semigroups where the polynomial is of the form \(x_1+x_2-x_3\).NEWLINENEWLINE This paper focuses on the case when the polynomial is of degree 1. It is said to be homogeneous if the free term is zero and the first part of the paper deals with these kinds of patterns. The authors give conditions on the coefficients as to when the set of semigroups admitting a given pattern is nonempty.NEWLINENEWLINE The second part deals with the nonhomogeneous case, more precisely when the constant term is a multiple of the multiplicity of the semigroup. Under some additional conditions on the polynomial (called strongly admissible) the set of semigroups satisfying them has the structure of an \(m\)-variety (or called nonhomogeneous Frobenius variety of multiplicity \(m\)) which allows for the description of the minimal sets of generators of the semigroups admitting these patterns.