Encuentro SINGACOM

We propose a measurement-free reconstruction circuit of messages in quantum codewords with erasures. Our reconstruction circuit has width $k+n-e$ and consists of $O(k(n-e))$ one- or two-qudit unitary gates when $k$-qudit quantum messages are reconstructed from $n$-qudit codewords with $e$ erasures. Details are available as arXiv:2505.18840. Joint work with my former student Shogo Chiwaki.

Earlier, there was defined and computed the Poincaré series in several contexts, all af them in the complex case. In two recent papers we give definitions of the Poincaré series of a collection of valuations in the real setting (i. e., on the ring of germs of functions on a real analytic variety), compute them for the case of one curve or divisorial valuation on the plane and discuss some of their properties. Recently we extend the computations to the case of other subfields of the complex numbers for one curve or divisorial valuation. The talk will focus on the case of a single valuation, pointing out the differences with the well-known case of branches over algebraically closed fields, as well as the qualitative difference between the semigroup and the Poincaré series. The results have been obtained in collaboration with S. Gusein-Zade and A. Campillo.

Let $\mathcal{C}$ be a positive integer cone. A $\mathcal{C}$-semigroup $S \subseteq \mathcal{C}$ is a cofinite affine semigroup. In this talk, inspired by the work presented in [1], we explore the computational aspects of $\mathcal{C}$-semigroups. We introduce new algorithms to compute $\mathcal{C}$-semigroups with prescribed invariants such as genus and Frobenius element. To support this approach, we define a new class of $\mathcal{C}$-semigroups, called $\mathcal{B}$-semigroups, which form the basis of our methods. Several examples and computational results will be presented.

It is a cojoint work with Jose Carlos Rosales and Alberto Vigneron-Tenorio.

[1] Rosales, J. C.; Tapia-Ramos, R.; Vigneron-Tenorio, A.: A computational Approach to the Study of Finite-Complement Submonids of an Affine Cone. Results Math. (2025) 80, no. 66.

In this talk we present new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods lead to the construction of new (not necessarily associative) division algebras and maximum rank distance (MRD) codes over both finite and infinite division rings. In particular, we construct new non-associative division algebras whose right nucleus is a central simple algebra having degree greater than 1. Over finite fields, we obtain new semifields and MRD codes for infinitely many choices of parameters. These families extend and contain many of the best previously known constructions.

In this talk, we introduce the open problem of determining whether a rational first integral exists for a foliation on the complex plane. Specifically, we present an algorithm that, assuming the genus (different from 1) of the rational first integral is known and a certain condition $C$ is not satisfied, provides an effective answer to the problem. In the case of polynomial rational first integrals of genus $g \neq 1$, the situation is completely resolved, as we can guarantee that condition $C$ is never fulfilled.

In the early 21st century, Campillo Delgado and Gusein-Zade discovered a fascinating connection between the Poincaré series of certain semigroups appearing in geometric contexts and the Alexander polynomial of some knots and links. In short, they show that there exist several cases where these two completely different invariants coincide up to a certain factor. It is then natural to ask to what extent one can naturally assign to any link a semigroup for which its Poincaré series is related to the Alexander polynomial of the link. In this context, in 2018 S. Wang has proposed the study of this situation in the case of L–space knots, which is an important family of knots with simplest Heegaard Floer homology.

In this talk, I will present a step forward in the approach to this problem. First, I will show a necessary condition for an L-space knot to have an Alexander polynomial equal, up to a factor, to the Hilbert–Poincaré series of a numerical semigroup. I will also provide some problems and conjectures related to this. Secondly, we will consider the reverse problem in a more general framework; not just L-space knots. From our point of view, the following questions are more natural: given a numerical semigroup, can we construct a knot whose Alexander polynomial equals the Hilbert–Poincaré series of the semigroup, up to a factor? If so, how many non-isotopic knots can be constructed from the same semigroup? Are any of these knots an L-space knot? I will also present some partial answers to those questions obtained in joint work with A. Olivares-Fernández.

The Nagata conjecture arises as a result of the negative answer to the 14th Hilbert problem [5]. Despite many efforts this conjecture remains open after more than 60 years. There exist several reformulations using interesting objects in Algebraic Geometry as Seshadri constants, Mori cone and irrational nef divisors. Recently, a valuative Nagata conjecture has been stated in [2,4]. This conjecture involves a (real) plane valuation $\nu$ of the projective plane and a constant named Seshadri-type constant, denoted $\hat{\mu}(\nu)$ and introduced in [1]. In addition, the valuative Nagata conjecture implies the Nagata conjecture and asymptotic evidences of its trueness in some particular cases have been proved in [4]. In this talk we introduce a natural Seshadri constant for a smooth projective surface $S$, a nef divisor on $S$ and a divisorial plane valuation of $S$ and we show some of their properties. These results will help us to give several equivalent geometrical statements to the valuative Nagata conjecture for divisorial plane valuations.

This talk is based on a joint work [3] with C. Galindo, F. Monserrat and J.-J. Moyano-Fernández.

[1] S. Boucksom, A. Küronya, C. Maclean, T. Szemberg , Vanishing sequences and Okounkov bodies, Math. Ann., 361 (2015), 811-834.

[2] M. Dumnicki, B. Harbourne, A. Küronya, J. Roé, T. Szemberg, Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture, Comm. Anal. Geom., 25 (2017) 125-161.

[3] C. Galindo, F. Monserrat, C.-J. Moreno-Ávila, J.-J. Moyano-Fernández, On the valuative Nagata conjecture, Res. Math. Sci. 12, 18 (2025).

[4] C. Galindo, F. Monserrat, J. J. Moyano-Fernández, Minimal plane valuations, J. Alg. Geom., 27 (2018), 751-783.

[5] M. Nagata. On the 14-th problem of Hilbert. Amer. J. Math., 81 (1959), 766--772.

We introduce homothetic-BCH codes. These are a family of $q^2$-ary classical codes $\mathcal{C}$ of length $\lambda n_1$, where $\lambda$ and $n_1$ are suitable positive integers such that the punctured code $\mathcal{B}$ of $\mathcal{C}$ in the last $\lambda n_1 -n_1$ coordinates is a narrow-sense BCH code of length $n_1$. We prove that whenever $\mathcal{B}$ is Hermitian self-orthogonal, so is $\mathcal{C}$. As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths than cannot be reached by BCH codes. With this procedure we get new quantum codes according to Grassl's table codetables.de. To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.

The set of normalized ideals of a numerical semigroup is the set of ideals whose minumum is zero. This set is a finite commutative monoid, and its idempotent elements are the oversemigroups of the numerical semigroup.

This set is a poset with respect to inclusion and also with respect to the order induced by addition (divisibility order). We will present some basic properties of these posets and some open problems.

In this talk we present a full description of an algebraic object naturally attached to the sky of a sequence of blow-ups, its Chow ring $A^{\bullet}(Z_{s})$, in two contexts: sequences of point blow-ups and sequences of point and rational curve blow-ups. This allows us to explore the fundamental differences arising in these two geometrical contexts as well as to investigate some combinatorial objects associated to this algebraic object that could be useful for computing invariants associated to different resolutions.