Towards a better understanding of the semigroup tree

Abstract

In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed in [2]. These regularites admit two different types of behavior and in this work we investigate which of the two types takes place in particular for well-known classes of semigroups. Also we study the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof. We conclude with some thoughts that show how this study of the semigroup tree may help in solving the conjecture of Fibonacci-like behavior of the number of semigroups with given genus.

Improved bounds on the number of numerical semigroups of a given genus

“…Similarly, rule (7) shows that the coefficient of t g+1 in K 0 (v, t) gets a contribution of v from the node labeled (2) when g = 1, a contribution of v 0 + v 2 from the node labeled (3) when g = 2, a contribution of v 0 from each node at level g labeled (e) with e ≥ 4 (there is one such node for each g ≥ 3), and a contribution of v from the node labeled (4) when g = 3. All these contributions yield the following equation for K 0 (v, t):…”

“…In order to translate the rules (13) into functional equations, it will be convenient to separate the terms in J(u, v, t) according to the parity of the exponent of v, so that J(u, v, t) = J e (u, v, t) + J o (u, v, t), with e and o standing for even and odd. Also, let F e (u, t) = u 4 t 3 1−u 2 t 2 and F o (u, t) = u 5 t 4 1−u 2 t 2 , so that F (u, t) = u 2 t +u 3 t 2 +F e (u, t)+F o (u, t). The coefficient of t g+1 in J e (u, v, t) gets a contribution of u λ/2 v λ +· · ·+u e v λ = u e+1 −u λ/2 u−1 v λ from each term u e v λ t g in J e (u, v, t), and a contribution of u λ/2 v λ + · · · + u λ−3 v λ + u λ−1 v λ = u λ−2 −u λ/2 u−1 v λ + u λ−1 v λ from each term u λ t g in F e (u, t).…”

Computation of numerical semigroups by means of seeds

“…Any semigroup of genus g ≥ 1 can be uniquely obtained from a semigroup Λ of genus g − 1 by removing a generator σ ≥ c(Λ). This allows one to arrange all numerical semigroups in an infinite tree, rooted at the trivial semigroup, such that the nodes at depth g are all semigroups of genus g. This tree was already introduced in [19,20] and later considered in [6,7,9]. The first nodes are shown in Figure 1, where each semigroup is represented by its non-zero elements up to the conductor.…”

“…The first values have been computed by Medeiros and Kakutani, Bras-Amorós [6], Delgado [10], and Fromentin and Hivert [13]. Several contributions [7,9,11,23,2,3,15,8,17] In this article we introduce the notion of seeds, as a generalization of the generators of a semigroup which are greater than the Frobenius number. This new concept allows us to explore the semigroup tree in an alternative efficient way, since the seeds of each descendant can be easily obtained from the seeds of its parent.…”

Singular rational curves with points of nearly-maximal weight

“…It is a rational curve of genus 8 with a unique singularity supported in P = (0 : 0 : 0 : 1), which is unibranch. The semigroup of P is S = {0, 3, 6, 9, 12, →}, and hence K = T. So the weight of P corresponds to (6) using the equality (5). On the other hand, one computes H 0 (ω C ) = dt/t 2 , dt/t 3 , dt/t 5 , dt/t 6 , dt/t 8 , dt/t 9 , dt/t 11 , dt/t 12 and from this point of view the weight of P corresponds to (7) using definition (2).…”