Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup

Abstract

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ N | ds ∈ T } and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is g + ⌈ (d−1)f 2 ⌉, where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of S and study the particular case when S is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups. MSC: 20M14; 13H10.Keywords Quotient of a numerical semigroup · Genus · Symmetric numerical semigroup · Almost symmetric semigroup · Frobenius number · d-symmetric semigroup · Numerical duplication.

On the genus of a quotient of a numerical semigroup

“…As mentioned in the introduction, the following corollary generalizes a result of Strazzanti [15]. Corollary 2.3.…”

“…are quasipolynomials in a of degree 2 (with periods 2 and 5, respectively) [15]. By specializing Theorem 1.3 to the case S = a, a + k , where k is a fixed positive integer that is coprime to a, we show that the genus of S/d is a quasipolynomial in a of degree 2 (see Corollary 2.3).…”

“…However, some special types of numerical semigroups have received a large amount of attention, and the invariants of these numerical semigroups are often well-understood. This includes numerical semigroups with small embedding dimensions [2,8,16], numerical semigroups generated by arithmetic progressions [6,10], and d-symmetric numerical semigroups [15]. For example, the following very fundamental result is due to Sylvester.…”

One-dimensional Gorenstein local rings with decreasing Hilbert function

“…The next proposition shows that, in the setting of the previous one, we only need to find an almost symmetric semigroup with decreasing Hilbert function. This is not true in general, for instance the numerical semigroup S = 30,35,42,47,108,110,113,118,122,127,134,139 is almost symmetric and its Hilbert function is…”

Counting numerical semigroups by genus and even gaps

“…For 2g ≥ 3γ (cf. [32]) we point out a quite useful parametrization, namely S γ (g) → S γ , S → S/2, which was introduced by Rosales et al [29] (see (2.3), [28], [17]). Thus Remark 2.11 shows the class of numerical semigroups we deal with in this paper; we do observe that these semigroups were already studied for example in [25] by using the concept of weight of semigroups.…”