Motivated by the way numerical semigroups arise from regular Markov chains, we will consider what we call the graphic structure of a numerical semigroup. Starting with the primitive digraph underlying a regular Markov chain, the collection of lengths of closed walks starting and ending at a vertex form a numerical semigroup provided we allow walks of length 0. The digraph in it¹s entirety gives rise to a collection of numerical semigroups connected by relative ideals. We are interested in the reverse viewpoint. Starting with a numerical semigroup, for what graphs can it correspond to a vertex? What is the minimal number of vertices such a graph can have? While this minimal number is certainly determined by the additive (i.e., algebraic) structure of the semigroup, it is not explicitly revealed by the usual representations of a semigroup such as enumeration, minimal generating set, or Apery sets. We introduce a new representation for numerical semigroups that reveals such graphic structure.

Joint work with Marco D’anna, Pedro Garcia-Sanchez and Vincenzo Micale.

Many properties of non irreducible singular algebraic curves can be studied through their value semigroup, which is a submonoid of $\mathbb{Z}^n$ closed under infimums, with a conductor and with particular combinatoric relations among its elements. Monoids fullfilling these three conditions are known as good semigroups.

In this talk we consider good semigroups independently from their algebraic counterpart, in a purely combinatoric setting. We define the concept of good system of generators, and we show that minimal good systems of generators are unique.

Semigroups of values of curve singularities satisfy certain axioms defining the (larger) class of good semigroups. We introduce a definition of quasihomogeneous good semigroups corresponding to the quasihomogeneity of curve singularities. A quasihomogeneous semigroup can be recovered from its components which are numerical semigroups. We give an additional condition which allows to construct from a quasihomogeneous semigroup $S$ a quasihomogeneous curve singularity having $S$ as semigroup of values.

We will describe the behaviour of Poincaré series associated with multi-index filtrations and value semigroups of curve singularities—not necessarily complex—with regard to the property of forgetting variables, i.e., by making subsets of variables of the series to be one.

Joint work with Pedro A. García-Sánchez and Benjamín A. Heredia

We compute the first and second Feng-Rao distance for all the elements in an Arf numerical semigroup that are greater than or equal to the conductor of such a semigroup. This provides a lower bound for the minimum distance and the second Hamming weight,respectively, for one point AG codes whose Weierstrass semigroup is Arf. In particular, we can obtain both Feng-Rao distances for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, we compute the second Feng-Rao number, and provide some examples and comparisons with previous results on this topic. These calculations rely on Apéry sets, and thus several results concerning Apéry sets of Arf semigroups are presented.

Joint work with Kasper Halbak Christensen

A classical research problem asks: Given the class of algebraic function fields over ${\mathbb{F}}_q$ and of genus $g$, what is the maximal number $N_q(g)$ of rational places. Running through all possible numerical semigroups $\Lambda$ with $g$ gaps and estimating for each semigroup what is the maximal number of rational places of a function field, should it have $\Lambda$ as a Weierstrass semigroup, partially answers the mentioned problem. Let $\Lambda=\langle w_1, \ldots ,w_m\rangle$ and define a monomial function $w(X_1^{i_1} \cdots X_m^{i_m})=i_1w_1+\cdots +i_mw_m$. Using methods from order domain theory we translate the above agenda into the following tasks:

1. Determine a Gröbner basis for the ideal $\langle \vec{X}^{\vec{\alpha}}-\vec{X}^{\vec{\beta}} \mid w(\vec{X}^{\vec{\alpha}})=w(\vec{X}^{\vec{\beta}})\rangle$ with respect to a clever weighted degree monomial ordering.
2. Add extra terms of lower weight to the above binomials (systematically or by random) and check if it is still a Gröbner basis.
3. In case of a yes, determine and count the affine roots.
4. Check if the curve is singular or non-singular.

Joint work with Arkadiusz Płoski.

We prove an intersection formula for two plane branches, in arbitrary characteristic, in terms of their semigroups and key polynomials.

In [vdK], van der Kulk proved a theorem on polynomial automorphisms of the plane generalizing a previous result of Jung [J] to the case of arbitrary characteristic. The proof of van der Kulk is based on a lemma on the intersection multiplicity of branches ([Lemma on page 36, vdK]) proved using the Hamburger-Noether expansions.

As application of our result we prove here a property of intersection multiplicities of branches which implies Van der Kulk’s lemma (see [P] for char $\mathbf{K=0}$).

Finally, as a consequence of our result we give a new proof of Bayer’s theorem (see [Theorem 5, B]) on the intersection multiplicities of two branches.

References

[B] Bayer, V. Semigroup of two irreducible algebroid plane curves, Manuscripta Mathematica, 39 (1985), 207-241.

[J] Jung, H. W. E. Uber ganze birationale Transformationen der Ebene. J. reine angew. Math. 184 (1942). 161-174.

[P] Płoski, A. B’ezout’s theorem for affine curves with one branch at infinity. Universitatis Iagellonicae Acta Matethematica, Fasciculus XXVIII (1991), 77-80.

[vDK] van der Kulk, W. On polynomial rings intwo variables. Nieuw Arch. Wiskunde (3) 1, (1953) 33-41.

Let S be a numerical semigroup with multiplicity m and conductor c. Let P denote the set of primitive elements of S and L the subset of elements of S which are strictly smaller than c. Herbert Wilf conjectured in 1978 that card(P)card(L) should always be greater than or equal to c. Wilf’s conjecture is known to hold in a number of cases, in particular when m is smaller than or equal to 8. We shall associate a certain graph G to S and shall use it to verify Wilf’s conjecture in some new cases, in particular when m is smaller than or equal to 12.

Joint work with D. Marín-Aragón and A. Vigneron-Tenorio.

Let $\mathcal{C}\subset \mathbb{Q}^p$ be a rational cone. An affine semigroup $S\subset \mathcal{C}$ is a $\mathcal{C}$-semigroup whenever $(\mathcal{C}\setminus S)\cap \mathbb{N}^p$ has only a finite number of elements.

In this work, we study the tree of $\mathcal{C}$-semigroups, give a method to generate it and study the $\mathcal{C}$-semigroups with minimal embedding dimension.We extend Wilf’s conjecture for numerical semigroups to $\mathcal{C}$-semigroups and give some families of $\mathcal{C}$-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for $\mathcal{C}$-semigroups.

Partially supported by MTM2014-55367-P, MTM2015-65764-C3-1-P (MINECO/FEDER, UE) and Junta de Andalucía group FQM-366.

References

[BM] Bras-Amorós, M., Fibonacci-like behavior of the number of numerical semigroups of a given genus, Semigroup Forum 76 (2008), no. 2, 379-384.

[FPU] Failla G.; Peterson C.; Utano, R., Algorithms and basic asymptotics for generalized numerical semigroups in $\mathbb{N}^p$, Semigroup Forum (2016) 92, 460-473.

[FF] Fromentin, J.; Florent H., Exploring the tree of numerical semigroups, Math. Comp. 85 (2016), no. 301, 2553–2568.

[K] Kaplan, N., Counting numerical semigroups by genus and some cases of a question of Wilf, J. Pure Appl. Algebra 216 (2012), no. 5, 1016-1032.

[M] Moscariello, A.; Sammartano, A., On a conjecture by Wilf about the Frobenius number, Math. Z. 280 (2015), no. 1-2, 47-53.

In 2008 Brás-Amoros conjectured that that the sequence $(n_{g})\scriptsize{g}$, where $n_g$ is the number of numerical semigroups of genus $g$, behaves like the Fibonacci sequence.

Counting numerical semigroups by genus became then a popular theme and the conjectured behavior of the sequence $(n_g)\scriptsize{g}$ resisted as a conjecture only for 5 years. It became a theorem in a work of Zhai, who also proved other related results, namely that the number of numerical semigroups of a given genus whose conductor is bigger that the triple of the multiplicity is negligible when asymptotically compared to the total number of semigroups of that same genus. Possible counter-examples to Wilf’s conjecture are among these semigroups whose conductor is ``large’’ when compared to the multiplicity, as proved by Eliahou. This suggests that if a counter-example to Wilf’s conjecture exists, it will possibly be hard to find.

We intend to survey the vast literature on the theme, mention some of the techniques used, refer some related existing conjectures and mention some new or unexplored approaches.

Joint work with D. Llena.

As far as we know, known implemented algorithms can’t compute denumerants for medium (hundreds of digits) or medium–large (thousands of digits) input data in a reasonable time cost on an ordinary computer. In this work we propose an algorithm that covers this gap for numerical $3$–semigroups.

Given $n_1,n_2,n_3\in{{\mathbb N}}$ with $1\leq n_1<n_2<n_3$ and $\gcd(n_1,n_2,n_3)=1$, the numerical $3$–semigroup $T={\langle{n_1,n_2,n_3}\rangle}$ is defined by $T=\{x_1n_1+x_2n_2+x_3n_3:~x_1,x_2,x_3\in{{\mathbb N}}\}$. A gap of $T$, $g$, is an element of $T^c={{\mathbb N}}\setminus T$. The Frobenius number is the maximum element in $T^c$, denoted by ${\mathfrak{f}}(a,b,c)$. Given $m\in T$, the set of factorizations of $m$ in $T$ is denoted by ${\mathcal{F}}(m,T)=\{(x_1,x_2,x_3)\in{{\mathbb N}}^3:~x_1n_1+x_2n_2+x_3n_3=m\}$. The denumerant of $m$ in $T$, ${\mathrm{d}}(m,T)$, is the cardinality of the set ${\mathcal{F}}(m,T)$.

Popoviciu in 1953 [P] found a semi–closed expression (this expression only requires $O(\log\max\{p,q\})$ arithmetic operations to be applied) of ${\mathrm{d}}(m,{\langle{p,q}\rangle})$ for $n=2$

$${\mathrm{d}}(m,{\langle{p,q}\rangle})=\frac{m+pf(m)+qg(m)}{{\mathrm{lcm}}(p,q)}-1,$$

where $f(m)\equiv-mp^{-1}{\!\!\pmod{q}}$ with $1\leq f(m)\leq q$ and $g(m)\equiv-mq^{-1}{\!\!\pmod{p}}$ with $1\leq g(m)\leq p$. This expression is efficient enough to accept large input data, with time cost $O(\log q)$ in the worst case.

No similar efficient semi–closed expressions are known for $k\geq3$, however there are some known numerical algorithms to find the set of factorizations ${\mathcal{F}}(m,T)$ in the general case. Unfortunately, as far as we know, usual computer algebra systems have implemented no command for denumerant. Thus, the calculation of denumerant turns to be a time consuming task. Taking for instance, $n_1=7^k$, $n_2=11^k$, $n_2={\mathfrak{f}}(7^k,11^k)$, $P_k=n_1n_2n_3$, $S_k=n_1+n_2+n_3$ and $m_k=P_k-S_k-k$, we obtain the figures of the following table for $T_k={\langle{n_1,n_2,n_3}\rangle}$ and ${\mathrm{d}}(m_k,T_k)$.

$k$	$m_k$	${\mathrm{d}}(m_k,T_k)$	Mathematica 8	Sage 7.3	GAP 4.8.3
1	4465	2232	0.011311	0.019617	0.009835
2	34139180	17069589	177.318173	535.270590	6.100101

Clearly, popular CAS programs cannot manage medium (hundreds of digits) or medium–large (thousands digits) inputs by using only their standard commands (the commands for computing the denumerant ${\mathrm{d}}(m,{\langle{a,b,c}\rangle})$ are Length[FrobeniusSolve[{a,b,c},m]] for Mathematica 8, WeightedIntegerVectors(m,[a,b,c]).cardinality() for Sage 7.3 and NrRestrictedPartitions(m,[a,b,c]) for GAP 4.8.3).

Popoviciu [P, page 27] gave an $O(c\log c)$ algorithm, in the worst case, for computing ${\mathrm{d}}(m,T)$ when $\{a,b,c\}$ are pairwise coprime numbers (pcn). Lisoněk in 1995 [L, page 230] gave an $O(ab\log b)$ algorithm, in the worst case for pcn (this time cost can be reduced to $O(ab)$ provided that a number of $\max\{O(a^2b^2), O(abc)\}$ precomputed values, related to $T$, can be stored in the computer memory for later usage). Brown, Chou and Shiue in 2003 [BCS, page 199] gave an $O(ab\log c)$ algorithm, in the worst case. This last work also contains interesting results on denumerants that can be taken into account for numerical calculations. In particular, they give a simple process to obviate the condition for $\{n_1,n_2,n_3\}$ to be pairwise coprime numbers. We refer these algorithms as P, L  and BCS, respectively. Notice that the speed of Algorithm P  versus Algorithm L depends on the ratio $\frac{c\log c}{ab\log b}$.

Algorithms P, L  and BCS  give the denumerants of folowing table significantly faster. A non-compiled Sage 7.3 implementation of them give the figures in the following (using the same processor; time in seconds).

$k$	$m_k$	${\mathrm{d}}(m_k,T_k)$	P	L	BCS
1	4465	2232	0.003994	0.004331	0.011661
2	34139180	17069589	0.083913	0.138889	0.920211
3	207657687311	103828843654	5.063864	9.495915	63.647251

Nonetheless, these algorithms don’t reach the necessary efficiency for managing medium or medium–large input. The goal of this work is to provide a reasonably efficient new algorithm which allows such kind of inputs when working on ordinary computers.

Our algorithm has a theoretical time cost of $O(b)$, in the worst case. However, numerical evidences suggest a lower cost. It is based on a semi-closed denumerant expression given in [AG]. As an instance, the time for $k=10^3$ is $0.009836$ seconds and for $k=10^5$ is $2.110464$ seconds.

References

[AG] F. Aguiló-Gost and P.A. García-Sánchez, Factoring in embedding dimension three numerical semigroups, Electron. J. Comb., 17 (2010), no. 1, R#138, 21 pp.

[BCS] Brown, Chou and Shiue, On the partition function of a finite set, Australasian J. Combin. 27 (2003), 193–204.

[L] P. Lisoněk, Denumerants and their approximations, J. Combin. Math. Combin. Comput. 18 (1995) 225–232.

[P] T. Popoviciu, Asupra unei probleme de patitie a numerelor, Acad. Republicii Populare Romane, Filiala Cluj, Studii si cercetari stiintifice 4 (1953) 7–58.

[RA] J.L. Ramírez Alfonsín, The Diophantine Frobenius Problem. Oxford Univ. Press (2005) Oxford. ISBN 0-19-856820-7 978-0-19-856820-9.

[RGS] Rosales, J. C. and García-Sánchez, P. A., Numerical semigroups. Developments in Mathematics, 20. Springer (2009) New York, ISBN: 978-1-4419-0159-0.

[^1]: This expression only requires $O(\log\max\{p,q\})$ arithmetic operations to be applied.

Joint work with J. I. García-García and D. Marín-Aragón.\

In this talk, following the work of S.T. Chapman, P.A. García-Sánchez, D. Llena and J. Marshall, Elements in a Numerical Semigroup with Factorizations of the Same Length, Canad. Math. Bull. 54(2011), 39-43, we show some bounds to calculate elements in a numerical semigroup with factorizations of the same length. We will also show how to improve the computation of Delta set of a numerical semigroup. That is, the set of distances between consecutive lengths for different factorizations of an element.

Partially supported by MTM2014-55367-P (MINECO/FEDER, UE) and Junta de Andalucía group FQM-366.

${\mathbb{R}}$-moulds of numerical semigroups are defined as increasing sequences of real numbers whose discretizations may give numerical semigroups. The ideal sequence of musical harmonics is an ${\mathbb R}$-mould and discretizing it is equivalent to defining equal temperaments. The number of equal parts of the octave in an equal temperament corresponds to the multiplicity of the related numerical semigroup.

Analyzing the sequence of musical harmonics two important properties of moulds are derived, those of being pythagorean and fractal. It is demonstrated that, up to normalization, there is only one pythagorean mould and one non-bisectional fractal mould. Furthermore, it is shown that the unique non-bisectional fractal mould is given by the golden ratio.

The case of half-closed cylindrical pipes imposes to the sequence of musical harmonics one third property, the so-called even-filterability property.

Joint work with J. Herzog and T. Hibi.

Nearly Gorenstein rings (or semigroups) have been recently introduced as a class of rings (resp. semigroups) which are close to being Gorenstein (respectively symmetric). For a numerical semigroup H, its canonical trace tr(H) is the ideal defined as the sum of the canonical and the anti-canonical ideals of H. A remarkable fact is that H is symmetric precisely when tr(H)=H. We say that H is nearly Gorenstein whenever tr(H) contains the generators of H. We show that this class contains the almost symmetric semigroups introduced by Barucci and Froberg and we classify the 3-generated ones.

To measure how far H is from being symmetric we let res(H) count the elements in H which are outside its canonical trace ideal. We provide bounds for res(H), and an exact formula when H is 3-generated. We give examples and discuss open questions regarding this new invariant and this new class of semigroups.

Joint work with S. Goto, S. Kumashiro, and N. Matsuoka.

The destination of this research is to find a good notion of Cohen-Macaulay local rings of positive dimension which naturally generalizes Gorenstein local rings. In dimension one, the research has started from the works of V. Barucci-R. Fröberg [BF] and S. Goto-N. Matsuoka-T. T. Phuong [GMP]. In [BF] Barucci and Fröberg introduced the notion of almost Gorenstein ring in the case where the local rings are one-dimensional and analytically unramified. They explored also numerical semigroup rings over fields and developed a beautiful theory. In [GMP] the authors extended the notion given by [BF] to arbitrary one-dimensional Cohen-Macaulay local rings and showed that their new definition works well to analyze almost Gorenstein rings which are analytically ramified. Our aim is to discover a good candidate for natural generalization of almost Gorenstein rings.

The present research has been strongly inspired by the researches of Sally modules [Section 4, GGHV] and [V]. We define 2-AGL rings in dimension one in terms of the rank of Sally modules of canonical ideals and the basic theory is developed. We also explore the case where the base rings are numerical semigroup rings over fields.

The author was partially supported by the International Research Supporting Program of Meiji University.

References

[BF] V. Barucci and R. Fröberg, One-dimensional almost Gorenstein rings, J. Algebra, 188 (1997), 418–442.

[GGHV] L. Ghezzi, S. Goto, J. Hong, and W. V. Vasconcelos, Invariants of Cohen-Macaulay rings associated to their canonical ideals, arXiv:1701.05592v1.

[GMP] S. Goto, N. Matsuoka, and T. T. Phuong, Almost Gorenstein rings, J. Algebra, 379 (2013), 355-381.

[V] V. W. Vasconcelos, The Sally modules of ideals: a survey, Preprint 2017, arXiv:1612.06261v1.

Given a numerical semigroup $S$ and a field $K$, we can construct the numerical semigroup ring $K[S]\subseteq K[x]$, consisting of polynomials with coefficients in $K$ where the only powers of $x$ permitted are those integers that appear in $S$. These integral domains have nonunique factorization and their factorization properties have been studied extensively. We will give an update on what is known about factorization in these rings, focusing in particular on the elasticity. Given a nonzero nonunit $a$, the elasticity $\rho(a)$ is the length of a longest factorization of $a$, divided by the shortest factorization length of $a$. The elasticity of the ring $R$ is $\displaystyle{\rho(R)=\sup_{a\in R} \rho(a)}$. We will discuss the value of $\rho(K[S])$, the distribution of elasticities of elements in $K[S]$, and how these questions relate to the analogous answers for $S$ itself. We will also discuss how the class group of $K[S]$ can be described in terms of the numerical monoid $S$.

Joint work with Marco D’Anna and Raheleh Jafari.

Let $S$ be a numerical semigroup, $E$ be a proper ideal of $S$ and $b \in S$ an odd integer. The numerical duplication $S \! \Join^b \! E$ of $S$ with respect to $E$ and $b$ is a numerical semigroup which arises in commutative algebra. This construction has had interesting applications both in numerical semigroup theory, e.g. regarding the doubles of a numerical semigroup, and in commutative algebra; for instance, the study of the numerical duplication and its associated graded ring led to construct the first known examples of one-dimensional Gorenstein local rings whose Hilbert function is decreasing (in some level). In this talk we are interested to deepen the knowledge of the associated graded ring of $S \! \Join^b \! E$, in particular studying its Cohen-Macaulayness, Gorensteinness, and other similar properties.

Star operations are closure operations that are classically defined on the set of fractional ideals of an integral domain; their definition can be extended in a natural way to numerical semigroups. The natural order on the set $\mathrm{Star}(S)$ of star operations on a numerical semigroup $S$ induces an order on the set $\mathcal{G}_0(S)$ composed of the (classes of) non-divisorial ideals of $S$, and there is a strong link between $\mathrm{Star}(S)$ and the set of the antichains of $\mathcal{G}_0(S)$. In this talk, we study the order on $\mathcal{G}_0(S)$, especially regarding ways of embedding it into $\mathbb{N}^n$, for some (possibly small) integer $n$.

Joint work with S. Goto, N. Matsuoka, and H. L. Truong.

Let $H = \left<a_1, a_2, \ldots , a_n\right>$ be a numerical semigroup and $R=k[H] = k[t^{a_1}, t^{a_2}, \ldots , t^{a_n}]\subseteq k[t]$ be the numerical semigroup ring of $H$ over a field $k$. The defining ideal $I$ of $R$ is defined by the kernel of the ring map $\varphi: k[x_1, x_2, \ldots , x_n] \to R$ such that $\varphi(x_i) = t^{a_i}$. The problem of exploring structure of $I$ is classical and important in Commutative Algebra. When $n=3$, J. Herzog completely solved this problem. However, even in the case $n=4$, there are some partial answers, for example, when $H$ is symmetric by H. Bresinsky \cite{B} and when $H$ is almost symmetric of type 2 by J. Komeda \cite{K}. In the present research, we explore the structure of $I$ when any pseudo-Frobenius number of $H$ is a multiple of some fixed integer $\alpha > 0$. The most important advantage of our result is that we don’t need any assumption about the number $n$. We then have the minimal free resolution of $R$ also. We use RF-matrices defined by A. Moscariello \cite{M} and the Eagon-Northcott complexes \cite{EN}.

The author was partially supported by the International Research Supporting Program of Meiji University.

References

[B] H. Bresinsky, Symmetric semigroups of integers generated by 4 elements, Manuscripta Math., 17 (1975), 205-219.

[EN] J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated to them, Proc. Roy. Soc. Ser. A, 269 (1962), 188-204.

[H] J. Herzog, Generators and relations of Abelian semigroups and semigroup rings, Manuscripta Math., 3 (1970) 175-193.

[K] J. Komeda, On the existence of Weierstrass points with a certain semigroup generated by 4 elements, Tsukuba J. Math. 6 (1982), 237–270.

[M] A. Moscariello, On the type of an almost Gorenstein monomial curve, J. Algebra, {\bf 456} (2016), 266–277.