Binomial ideal of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup minimally generated by \(P(S) = \{n_1, \ldots, n_e\}\) and \(\mathbb{K}\) a field. Let us consider \(\mathbb{K}[x_1, \ldots, x_e]\) the polynomial ring with indeterminate \(x_1, \ldots, x_e\) and \(\mathbb{K}[t]\) the polynomial ring with indeterminate \(t\). Let us also consider the ring homomorphism
\[ \varphi: \mathbb{K}[x_1, \ldots, x_e] \to \mathbb{K}[t], ~~ \varphi(x_i) = t^{n_i} ~~ \text{for all } i \in \{1, \ldots, e\}, \]
and extended linearly. It is defined the binomial ideal of \(S\) as the kernel of \(\varphi\), \(\ker(\varphi)\). The image of the homomorphism is the semigroup ring associated to \(S\). Since \(\mathbb{K}\) is a filed, it is well known that \(\mathbb{K}[x_1, \ldots, x_e]\) is a noetherian ring, from which it is deduced that \(\ker(\varphi)\) is finitely generated.
Moreover, it can be proven that \(\rho = \{(\alpha_1, \beta_1), \ldots, (\alpha_n, \beta_n)\}\) with \(\alpha_1, \ldots, \alpha_n, \beta_1, \ldots, \beta_n \in \mathbb{N}^e\) is a minimal presentation of \(S\) if, and only if,
\[ \ker(\varphi) = \langle X^{\alpha_1} - X^{\beta_1}, \ldots, X^{\alpha_n} - X^{\beta_n} \rangle, \]
where \(\alpha_i = (\alpha_{i,1}, \ldots, \alpha_{i,e})\), \(X^{\alpha_i} = X_1^{\alpha_{i,1}} \cdots X_e^{\alpha_{i,e}}\) and \(\beta_i = (\beta_{i,1}, \ldots, \beta_{i,e})\), \(X^{\beta_i} = X_1^{\beta_{i,1}} \cdots X_e^{\beta_{i,e}}\).
Examples
\(\circ\) Let \(S = \langle 15, 24, 25, 36 \rangle\). Since \(S\) is a gluing of \(S_1 = \langle 3, 5 \rangle\) and \(S_2 = \langle 2, 3 \rangle\) with \(\lambda = 12\) and \(\mu = 5\) and \(1 \cdot 24 + 1 \cdot 36 = 4 \cdot 15\), it is deduced that a minimal presentation of \(S\) is
\[ \rho = \{[ (0, 0, 0, 2), (0, 3, 0, 0) ], [ (0, 0, 3, 0), (1, 1, 0, 1) ], [ (0, 1, 0, 1), (4, 0, 0, 0 ) ] \}. \]
Then, the binomial ideal of \(S\) in \(\mathbb{Q}\) is
\[ \ker(\varphi) = \langle X_4^{2} - X_2^3, X_3^3 - X_1X_2X_4, X_2X_4 - X_1^4 \rangle. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 12, 21, 30, 39, 56, 64 \rangle\), in GAP:
gap> S := NumericalSemigroup(12, 21, 30, 39, 56, 64);
<Numerical semigroup with 6 generators>Given a numerical semigroup \(S\) and a field \(\mathbb{K}\), the function BinomialIdealOfNumericalSemigroup returns the binomial ideal of \(S\) in \(\mathbb{K}\). If no field is given, then it is considered the field of rational numbers. The function GeneratorsOfTwoSidedIdeal returns a list with the generators of the ideal.
gap> I := BinomialIdealOfNumericalSemigroup(S);
<two-sided ideal in Rationals[x_1,x_2,x_3,x_4,x_5,x_6], (9 generators)>
gap> GeneratorsOfTwoSidedIdeal(I);
[ -x_2^2*x_3*x_5+x_6^2, -x_3^4+x_5*x_6, -x_1^4*x_6+x_5^2, -x_1^3*x_2^2+x_4^2,
-x_1^4*x_2+x_3*x_4, -x_1^5+x_3^2, -x_1^5+x_2*x_4, -x_1*x_4+x_2*x_3, -x_1*x_3+x_2^2 ]References
https://gap-packages.github.io/ numericalsgps.