Degree asymptotics of the numerical semigroup tree
Degree asymptotics of the numerical semigroup tree
复制标题
数值半群树的渐近度
作者：
Evan M. O’Dorney
一个数值半群是非负整数的子集Λ，它在加法下是闭的，包含0，并且只省略了1/2个非负整数（称为Λ的间隙）。所有数值半群的集合可以用元素移除树来直观地表示，其中半群Λ的子半群是通过移除Λ中超过Λ的所有现有间隙的一个元素来形成的。一般来说，一个半群可能有很多子群，也可能没有子群，这使得我们很难理解在树的给定深度上半群的个数。我们研究了在深度g上具有h个子半群的半群个数的估计问题，证明了当g变大时，它在所有数值半群中的比例趋于φ − h − 2，其中φ是黄金比例。
A numerical semigroup is a subset Λ of the nonnegative integers that is closed under addition, contains 0, and omits only finitely many nonnegative integers (called the gaps of Λ). The collection of all numerical semigroups may be visually represented by a tree of element removals, in which the children of a semigroup Λ are formed by removing one element of Λ that exceeds all existing gaps of Λ. In general, a semigroup may have many children or none at all, making it difficult to understand the number of semigroups at a given depth on the tree. We investigate the problem of estimating the number of semigroups at depth g with h children, showing that as g becomes large, it tends to a proportion φ−h−2 of all numerical semigroups, where φ is the golden ratio.