Cyclotomic exponent sequence
Definition
Let \(S\) be a numerical semigroup, \(H_S(x)\) its Hilbert series and \(P_S(x) = (1-x) H_S(x)\). It can be proven that there exists unique integers \(e_j\) such that the formal identity
\[ P_S(x) = \prod_{j = 1}^{+ \infty} (1 - x^j)^{e_j}, \]
holds. It is defined the cyclotomic exponent sequence of \(S\), denoted by \(\mathbf{e}(S)\), as \(\mathbf{e}(S) = \{e_j\}_{j \ge 1}\).
It can be proven that if \(S\) is a complete intersection, then the cyclotomic exponent sequence is finite.
Examples
\(\circ\) Let \(S = \langle 4, 6, 9 \rangle\). With some computations (see Hilbert series of a numerical semigroup), it holds that
$$ H_S(x) = ,
$$
and then,
\[ P_S(x) = (1 - x)H_S(x) = x^{12}-x^{11}+x^8-x^7+x^6-x^5+x^4-x+1 \]
\[ = (1-x)(1-x^4)^{-1}(1-x^6)^{-1}(1-x^9)^{-1}(1-x^{12})(1-x^{18}). \]
Therefore, \(\mathbf{e}(S) = \{1,0,0,-1,0,-1,0,0,-1,0,0,1,0,0,0,0,0,1,0, \ldots\}\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 15, 37, 56, 64, 95, 121 \rangle\), in GAP:
gap> S := NumericalSemigroup(15, 37, 56, 64, 95, 121);
<Numerical semigroup with 6 generators>Given a numerical semigroup \(S\) and a positive integer \(k\), the function CyclotomicExponentSequence returns a list with the first \(k\) elements of the cyclotomic exponent sequence of \(S\).
gap> CyclotomicExponentSequence(S, 40);
[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, -1, 0, 0, 0 ]References
https://gap-packages.github.io/ numericalsgps.