Fibonacci-like growth of numerical semigroups of a given genus

We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if n _g is the number of numerical semigroups of genus g, we prove that $$\lim_{g \rightarrow \infty} n_g \varphi^{-g} = S $$ where $\varphi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio and S is a constant, resolving several related conjectures concerning the growth of n _g . In addition, we show that the proportion of numerical semigroups of genus g satisfying f<3m approaches 1 as g→∞, where m is the multiplicity and f is the Frobenius number.     

Bounds on the number of numerical semigroups of a given genus

Lower and upper bounds are given for the number n_g of numerical semigroups of genus g. The lower bound is the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers. In a previous work the sequence n_g is conjectured to behave asymptotically as the Fibonacci numbers. The lower bound proved in this work is related to the Fibonacci numbers and so the result seems to be in the direction to prove the conjecture. The method used is based on an accurate analysis of the tree of numerical semigroups and of the number of descendants of the descendants of each node depending on the number of descendants of the node itself.     

Constructing numerical semigroups of a given genus

Let n _g denote the number of numerical semigroups of genus g. Bras-Amorós conjectured that n _g possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree. We offer a new, simpler approach to counting numerical semigroups of a given genus. Our method gives direct constructions of families of numerical semigroups, without referring to the generators or the semigroup tree. In particular, we give an improved asymptotic lower bound for n _g .     

Counting numerical semigroups by genus and even gaps

Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_{\gamma }\left(g\right)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $\gamma$. We show that $N_{\gamma }\left(g\right)=N_{\gamma }\left(3\gamma \right)$ for $\gamma \le ?g∕3?$ and $N_{\gamma }\left(g\right)=0$ for $\gamma >?2g∕3?$; thus the question above is true provided that $N_{\gamma }(g+1)>N_{\gamma }\left(g\right)$ for $\gamma =?g∕3?+1,\dots ,?2g∕3?$. We also show that $N_{\gamma }\left(3\gamma \right)$ coincides with $f_{\gamma }$, the number introduced by Bras-Amorós (2012) in connection with semigroup-closed sets. Finally, the stronger possibility $f_{\gamma }～{\phi }^{2\gamma }$ arises being $\phi =(1+\sqrt{5})∕2$ the golden number.     

The set of numerical semigroups of a given genus

In this paper we present a new approach to construct the set of numerical semigroups with a fixed genus. Our methodology is based on the construction of the set of numerical semigroups with fixed Frobenius number and genus. An equivalence relation is given over this set and a tree structure is defined for each equivalence class. We also provide a more efficient algorithm based on the translation of a numerical semigroup to its so-called Kunz-coordinates vector.     

On the enumeration of the set of saturated numerical semigroups of a given genus

In this paper we present an algorithm for computing the set of saturated numerical semigroups of a given genus. We see how the set of saturated numerical semigroups can be arranged in a tree rooted in \(\mathbb{N}\) and we describe the sons of any vertex of this tree.     

Ring semigroups whose subsemigroups form a chain

A multiplicative semigroup S is called a ring semigroup if an addition may be defined on S so that (S,+,⋅) is a ring. Such semigroups have been well-studied in the literature (see Bell in Words, Languages and Combinatorics, pp. 24–31, World Scientific, Singapore, 1994; Jones in Semigroup Forum 47(1):1–6, 1993; Jones and Ligh in Semigroup Forum 17(2):163–173, 1979). In this note, we use Mihăilescu’s Theorem (formerly Catalan’s Conjecture) to characterize the ring semigroups whose subsemigroups containing 0 form a chain with respect to set inclusion.     

Every numerical semigroup is one half of a symmetric numerical semigroup

Let be a numerical semigroup. Then there exists a symmetric numerical semigroup such that .

     

Lattice paths with given number of turns and semimodules over numerical semigroups

Let Γ=〈α,β〉 be a numerical semigroup. In this article we consider several relations between the so-called Γ-semimodules and lattice paths from (0,α) to (β,0): we investigate isomorphism classes of Γ-semimodules as well as certain subsets of the set of gaps of Γ, and finally syzygies of Γ-semimodules. In particular we compute the number of Γ-semimodules which are isomorphic with their k-th syzygy for some k.     

Densities of maximal embedding dimension numerical semigroups

We define the density of a numerical semigroup and study the densities of all the maximal embedding dimension numerical semigroups with a fixed Frobenius number, as well as the possible Frobenius number for a fixed density. We also prove that for a given possible density, in the sense of Wilf’s conjecture, one can find a maximal embedding dimension numerical semigroup with that density.     

On the number of curves of genus 2 over a finite field

In this paper we compute the number of curves of genus 2 defined over a finite field k of odd characteristic up to isomorphisms defined over k; the even characteristic case is treated in an ongoing work (G. Cardona, E. Nart, J. Pujolàs, Curves of genus 2 over field of even characteristic, 2003, submitted for publication). To this end, we first give a parametrization of all points in , the moduli variety that classifies genus 2 curves up to isomorphism, defined over an arbitrary perfect field (of zero or odd characteristic) and corresponding to curves with non-trivial reduced group of automorphisms; we also give an explicit representative defined over that field for each of these points. Then, we use cohomological methods to compute the number of k-isomorphism classes for each point in .     

The Darmon-Dasgupta units over genus fields and the Shimura correspondence

We study a special case of the Gross-Stark conjecture (Gross, 1981 [Gr]), namely over genus fields. Based on the same idea we provide evidence of the rationality conjecture of the elliptic units for real quadratic fields over genus fields, which is a refinement of the Gross-Stark conjecture given by Darmon and Dasgupta (2006) [DD]. Then a relationship between these units and the Fourier coefficients of p-adic Eisenstein series of half-integral weight is explained.     

Semitransitive subsemigroups of the symmetric inverse semigroups

We initiate the study of semitransitive transformation semigroups. In the paper we describe the structure of semitransitive subsemigroups of the finite symmetric inverse semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups, and state the results on minimal transitive subsemigroups. The authors were supported in part by Ukrainian-Slovenian bilateral research grants from the Ministry of Education and Science, Ukraine, and the Research Agency of the Republic of Slovenia.     

Numerical semigroups with many gaps of the same parity

Let g _e (S) (respectively, g _o (S)) be the number of even (respectively, odd) gaps of a numerical semigroup S. In this work we study and characterize the numerical semigroups S that verify 2|g _e (S)−g _o (S)|+1∈S. As a consequence we will see that every numerical semigroup can be represented by means of a numerical semigroup with maximal embedding dimension with all its minimal generators odd. The first author is supported by the project MTM2007-62346 and FEDER funds. The authors want to thank P.A. García-Sánchez and the referee for their comments and suggestions.     

Uniform bounds on the number of rational points of a family of curves of genus 2

We exhibit a genus-2 curve defined over which admits two independent morphisms to a rank-1 elliptic curve defined over . We describe completely the set of -rational points of the curve and obtain a uniform bound on the number of -rational points of a rational specialization of the curve for a certain (possibly infinite) set of values . Furthermore, for this set of values we describe completely the set of -rational points of the curve . Finally, we show how these results can be strengthened assuming a height conjecture of Lang.