Kunz coordinates
Definition
Let \(S\) be a numerical semigroup, \(n \in S\) and let \(Ap(S,n)\) be the Apéry set of \(S\) in \(n\). It is well known that
\[ Ap(S,n) := \{w \in S ~ | ~ w - n \not \in S\} = \{w(0) = 0, w(1), \ldots, w(n-1)\}, \]
where \(w(k) \in S\) is the first element in \(S\) such that \(w(k) \equiv k ~ (mod ~ n)\) for all \(k \in \{0, \ldots, n-1\}\). Therefore, there exist \(x_1, \ldots, x_{n-1} \in \mathbb{N}\) such that
\[ w(k) = x_kn + k, ~~ \text{for all } ~ k \in \{0, \ldots, n-1\}. \]
It is defined the Kunz coordinates of \(S\) with respect to \(n\) as the tuple \((x_1, \ldots, x_{n-1}) \in \mathbb{N}^{n-1}\). If we consider \(n = m(S)\), where \(m(S)\) denotes the multiplicity of \(S\), it is said directly as the Kunz coordinates of \(S\).
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{0, 5, 7, 9, 10, 12, 14, \rightarrow\}\) and \(n = 5\). From the definition, \(Ap(S, 5) = \{0, 7, 9, 16, 18\}\). It holds that
\[ w(1) = 16 = 3 \cdot 5 + 1, ~ w(2) = 7 = 1 \cdot 5 + 2, \]
\[ w(3) = 18 = 3 \cdot 5 + 3, ~ w(4) = 9 = 1 \cdot 5 + 4, \]
obtaining that the tuple \((3, 1, 3, 1)\) is the Kunz coordinates of \(S\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 13, 42, 43, 72, 79 \rangle\), in GAP:
gap> S := NumericalSemigroup(13, 42, 43, 72, 79);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\) and an integer \(n \in S\), the functions KunzCoordinates and KunzCoordinatesOfNumericalSemigroup return the Kunz coordinates of \(S\) with respect to \(n\).
gap> KunzCoordinates(S, 42);
[ 1, 2, 3, 3, 3, 4, 2, 2, 3, 1, 2, 3, 0, 1, 2, 3, 3, 3, 4, 2, 2, 3, 1, 2, 3, 0, 1,
2, 3, 1, 2, 3, 2, 2, 3, 1, 1, 2, 0, 1, 2 ]
gap> KunzCoordinatesOfNumericalSemigroup(S, 42) = KunzCoordinates(S, 42);
trueIf we only give a numerical semigroup, the functions computes the Kunz coordinates of \(S\) with respect to its multiplicity.
gap> KunzCoordinates(S);
[ 6, 12, 3, 3, 9, 6, 5, 6, 9, 8, 8, 9 ]
gap> KunzCoordinatesOfNumericalSemigroup(S) = KunzCoordinates(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.