A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry. Most arguments of this type make use of a parametrization of numerical semigroups with fixed multiplicity m in terms of their m-Apéry sets, giving a representation called Kunz coordinates which obey a collection of inequalities defining the Kunz polyhedron. In this work, we introduce a new class of polyhedra describing numerical semigroups in terms of a truncated addition table of their positive sporadic elements. Applying a classical theorem of Ehrhart to slices of these polyhedra, we prove that the number of numerical semigroups with n sporadic elements and Frobenius number f is polynomial up to periodicity, or quasi-polynomial, as a function of f for fixed n. We also generalize this approach to higher dimensions to demonstrate quasi-polynomial growth of the number of affine semigroups with a fixed number of elements, and all gaps, contained in an integer dilation of a fixed polytope.

数值半群理论中的一个常见工具是将所需的半群类解释为有理多面体中的格点，以便利用多面体几何中的计算和枚举技术。这种类型的大多数参数都利用数值半群的参数化，该数值半群的重数为m ，以m -Apéry 集表示，给出称为 Kunz 坐标的表示，该表示遵循定义 Kunz 多面体的不等式集合。在这项工作中，我们引入了一类新的多面体，根据正零星元素的截断加法表来描述数值半群。将埃尔哈特经典定理应用于这些多面体的切片，我们证明了具有n的数值半群的数量零星元素和弗罗贝尼乌斯数f是周期性的多项式或拟多项式，作为固定n的f的函数。我们还将这种方法推广到更高的维度，以证明具有固定数量的元素和所有间隙的仿射半群数量的拟多项式增长，包含在固定多胞形的整数膨胀中。