Kunz polytope
Definition
Let \(m\) and \(g\) be positive integers. It is defined the Kunz polytope, denoted by \(K(m,g)\), as the system of equations
\[ \begin{cases} x_i \ge 1 & \text{for all } ~ i = 1, 2, \ldots, m-1, \\ x_i + x_j - x_{i+j} \ge 0 & \text{for all } ~ 1 \le i \le j \le m-1 ~ \text{and} ~ i + j \le m-1, \\ x_i + x_j - x_{i+j-m} \ge -1 & \text{for all } ~ 1 \le i \le j \le m-1 ~ \text{and} ~ i + j \ge m+1, \\ x_1 + x_2 + \cdots + x_{m-1} = g. \end{cases} \]
It can be proven that there is a one-to-one correspondence between the number of numerical semigroups with multiplicity \(m\) and genus \(g\) and the set of positive integers \((x_1, \ldots, x_{m-1})\) that are solutions of the Kunz polytope.
Examples
\(\circ\) Let us consider the numerical semigroup \(S = \langle 3, 7, 11 \rangle\), which has multiplicity \(m(S) = 3\) and genus \(g(S) = 5\). Let us see that the Kunz coordinates of \(S\) is a solution of the Kunz polytope \(K(3,5)\). The Apéry set of \(S\) with respect to \(m(S)\) is \(Ap(S,3) = \{0, 7, 11\}\) and \(7 = 2 \cdot 3 + 1, 11 = 3 \cdot 3 + 2\). Then, the Kunz coordinates of \(S\) is \((x_1, x_2) = (2,3)\). On the other hand, the Kunz polytope \(K(3, 5)\) is the following system of equations.
\[ \begin{cases} x_1 \ge 1, \\ x_2 \ge 1, \\ 2x_1 - x_2 \ge 0, \\ -x_1 + 2x_2 \ge -1, \\ x_1 + x_2 = 5. \end{cases} \]
We have that \(2 \ge 1, ~ 3 \ge 1, ~ 2 \cdot 2 - 3 = 1 \ge 0, ~-2 + 2 \cdot 3 = 4 \ge -1\) and \(2 + 3 = 5\), concluding that \((2,3)\) is an integer solution of \(K(3,5)\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Given a positive integer \(m\), the function KunzPolytope returns the matrix \((A | -b)\) such that \(Ax \ge b\) is the Kunz polytope removing the condition \(x_1 + \cdots + x_{m-1} = g\).
gap> L := KunzPolytope(5);
[ [ 1, 0, 0, 0, -1 ], [ 0, 1, 0, 0, -1 ], [ 0, 0, 1, 0, -1 ], [ 0, 0, 0, 1, -1 ],
[ 2, -1, 0, 0, 0 ], [ 1, 1, -1, 0, 0 ], [ 1, 0, 1, -1, 0 ], [ 0, 2, 0, -1, 0 ],
[ -1, 1, 0, 1, 1 ], [ -1, 0, 2, 0, 1 ], [ 0, -1, 1, 1, 1 ], [ 0, 0, -1, 2, 1 ] ]Therefore, the Kunz polytope \(K(m,g)\) would be
\[ \begin{pmatrix} 1 & 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & -1 \\ 2 & -1 & 0 & 0 & 0 \\ 1 & 1 & -1 & 0 & 0 \\ 1 & 0 & 1 & -1 & 0 \\ 0 & 2 & 0 & -1 & 0 \\ -1 & 1 & 0 & 1 & 1 \\ -1 & 0 & 2 & 0 & 1 \\ 0 & -1 & 1 & 1 & 1 \\ 0 & 0 & -1 & 2 & 1 \\ 1 & 1 & 1 & 1 & -g \\ \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ 1 \\ \end{pmatrix} \ge \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix}. \]
References
https://gap-packages.github.io/ numericalsgps.