Weierstrass weight
Definition
Let \(S\) be a numerical semigroup and let \(G(S) = \{l_1, \ldots, l_g\}\) be the set of gaps of \(S\) with \(l_1 < l_2 < \ldots < l_g\). It is defined the Weierstrass weight of \(S\), denoted by \(w(S)\), as
\[ w(S) = \sum_{i = 1}^g (l_i - i). \]
From the definition, since the least possible gap is \(1\), we have that \(i \le l_i\) for all \(i \in \{1, \ldots, g\}\), from which it follows that the Weierstrass weight is always non-negative and \(w(S) = 0\) if, and only if, \(S\) is a half-line.
Examples
\(\circ\) Let \(S = \langle 5, 9, 12 \rangle = \{0, 5, 9, 10, 12, 14, 15, 17, \rightarrow\}\). The set of gaps of \(S\) is
\[ G(S) = \{1, 2, 3, 4, 6, 7, 8, 11, 13, 16 \}, \]
and the genus of \(S\) is \(|G(S)| = 10\). Then, the Weierstrass weight of \(S\) is
\[ w(S) = \sum_{i = 1}^{10} (l_i - i) = \sum_{i = 1}^4 (l_i - i) + \sum_{i = 5}^7 (l_i - i) + (l_8-8) + (l_9-9) + (l_{10}-10) = 0 + 3 + 3 + 4 + 6 = 16. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 23, 82, 93, 94, 143 \rangle\), in GAP:
gap> S := NumericalSemigroup(23, 82, 93, 94, 143);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the function Weight returns the Weierstrass weight of \(S\).
gap> Weight(S);
5642References
https://gap-packages.github.io/ numericalsgps.