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Maria Bras-Amorós

Publications and source records attributed to Maria Bras-Amorós.

At least 19 recordsLinked to original sources
arXiv2026

Multiparameter counting of numerical semigroups: recurrences and leaf-discriminating trees

A numerical semigroup is a subset of the nonnegative integers, closed under addition and with finite complement. The size of the complement is its genus. The problem of counting semigroups by the largest gap got very important through the so-called Frobenius problem, first documented in 1884. Accordingly, the largest gap is called the Frobenius number. In the last two decades, counting by the genus has become a subject of even more intense study, mostly because of the non-solved conjectures on its monotonic and super-Fibonacci growth. We propose a new approach to counting semigroups by the Frobenius number and by the genus, by introducing two ad-hoc trees. Those are leaf-discriminating trees in the sense that their leaves correspond exactly to the objects we want to count and, so, exploring these trees is optimal. On the theoretical side, it is known that the number of semigroups of each genus grows asymptotically with the genus as the Fibonacci numbers and that the number of semigroups of each Frobenius number grows asymptotically as a two-step doubling sequence. We prove a formula for the number of numerical semigroups of each Frobenius number $F$, genus $g$, and multiplicity $m$ (first nonzero nongap), for $m\geq(F+1)/3$. It is known that asymptotically almost all semigroups satisfy this inequality. This formula gives a multiparameter exact version of the increasing behaviours just mentioned. On the computational side, we implemented a recursive descending algorithm based on the so-called seeds structure, trimming the general semigroup tree exactly at those nodes with no descendants with a given genus, in the first case, or with no descendants with a given Frobenius number, in the second case. We refined the parallelizing strategies and we overcame the previous limitation of the length of integers in the bitwise representation of the gap sequence and the seed sequence.

cs.DM↗
arXiv2026

Exploring the unleaved tree of numerical semigroups up to a given genus

We present a new algorithm to explore or count the numerical semigroups of a given genus which uses the unleaved version of the tree of numerical semigroups. In the unleaved tree there are no leaves rather than the ones at depth equal to the genus in consideration. For exploring the unleaved tree we present a new encoding system of a numerical semigroup given by the gcd of its left elements and its shrinking, that is, the semigroup generated by its left elements divided by their gcd. We show a method to determine the right generators and strong generators of a semigroup by means of the gcd and the shrinking encoding, as well as a method to encode a semigroup from the encoding of its parent or of its predecessor sibling. With the new algorithm we obtained $n_{76}=29028294421710227$ and $n_{77}=47008818196495180$.

math.CO↗
arXiv2026

The shape of a random numerical semigroup

We study statistical properties of random numerical semigroups of a given genus. We analyze the graph of a typical numerical semigroup, understood as a function from $\mathbb{N}$ to $\mathbb{N}$. If $S$ is a numerical semigroup of genus $g$, this leads us to consider the collection of points $\left(\frac{k-1}{g-1},\frac{a_k(S)}{g} \right)$ where $1 \le k \le g$ and $a_k(S)$ denotes the $k$th smallest nonzero element of $S$. We show that as $g \rightarrow \infty$, this set of points typically becomes closer to a union of two line segments. We prove analogous results for numerical semigroups ordered by Frobenius number.

math.CO↗
arXiv2026

Memory poisoning and secure multi-agent systems

Memory poisoning attacks for Agentic AI and multi-agent systems (MAS) have recently caught attention. It is partially due to the fact that Large Language Models (LLMs) facilitate the construction and deployment of agents. Different memory systems are being used nowadays in this context, including semantic, episodic, and short-term memory. This distinction between the different types of memory systems focuses mostly on their duration but also on their origin and their localization. It ranges from the short-term memory originated at the user's end localized in the different agents to the long-term consolidated memory localized in well established knowledge databases. In this paper, we first present the main types of memory systems, we then discuss the feasibility of memory poisoning attacks in these different types of memory systems, and we propose mitigation strategies. We review the already existing security solutions to mitigate some of the alleged attacks, and we discuss adapted solutions based on cryptography. We propose to implement local inference based on private knowledge retrieval as an example of mitigation strategy for memory poisoning for semantic memory. We also emphasize actual risks in relation to interactions between agents, which can cause memory poisoning. These latter risks are not so much studied in the literature and are difficult to formalize and solve. Thus, we contribute to the construction of agents that are secure by design.

cs.CR↗
arXiv2024

Greedy Sets and Greedy Numerical Semigroups

Motivated by the change-making problem, we extend the notion of greediness to sets of positive integers not containing the element $1$, and from there to numerical semigroups. We provide an algorithm to determine if a given set (not necessarily containing the number $1$) is greedy. We also give specific conditions for sets of cardinality three, and we prove that numerical semigroups generated by three consecutive integers are greedy.

math.CO↗
arXiv2024

Infinite chains in the tree of numerical semigroups

One major problem in the study of numerical semigroups is determining the growth of the semigroup tree. In the present work, infinite chains of numerical semigroups in the semigroup tree, firstly introduced by Bras-Amorós and Bulygin (Semigroup Forum, 79:561--574, 2009), are studied. Computational results show that these chains are rare, but without them the tree would not be infinite. It is proved that for each genus $g\geq 5$ there are more semigroups of that genus not belonging to infinite chains than semigroups belonging. Bras-Amorós and Bulygin (Semigroup Forum, 79:561--574, 2009) presented a characterization of the semigroups that belong to infinite chains in terms of the coprimality of the left elements of the semigroup as well as a result on the cardinality of the set of infinite chains to which a numerical semigroup belongs in terms of the primality of the greatest common divisor of these left elements. We revisit these results and fix an imprecision on the cardinality of the set of infinite chains to which a semigroup belongs in the case when the greatest common divisor of the left elements is a prime number. We then look at infinite chains in subtrees with fixed multiplicity. When the multiplicity is a prime number there is only one infinite chain in the tree of semigroups with such multiplicity. When the multiplicity is $4$ or $6$ we prove a self-replication behavior in the subtree and prove a formula for the number of semigroups in infinite chains of a given genus and multiplicity $4$ and $6$, respectively.

cs.DM↗
arXiv2024

Self-orthogonal flags of codes and translation of flags of algebraic geometry codes

A flag $C_0 \subsetneq C_1 \cdots \subsetneq C_s \subsetneq {\mathbb F}_q^n $ of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy $C_{i}^\perp=C_{s-i}$, and it is said to satisfy the isometry-dual property with respect to an isometry vector ${\bf x}$ if $C_i^\perp={\bf x} C_{s-i}$ for $i=1, \dots, s$. We characterize complete (i.e. $s=n$) flags with the isometry-dual property by means of the existence of a word with non-zero coordinates in a certain linear subspace of ${\mathbb F}_q^n$. For flags of algebraic geometry (AG) codes we prove a so-called translation property of isometry-dual flags and give a construction of complete self-orthogonal flags, providing examples of self-orthogonal flags over some maximal function fields. At the end we characterize the divisors giving the isometry-dual property and the related isometry vectors showing that for each function field there is only a finite number of isometry vectors and that they are related by cyclic repetitions.

cs.IT↗
arXiv2024

Rarity of the infinite chains in the tree of numerical semigroups

We prove that, for each fixed genus, the portion of semigroups of that genus belonging to infinite chains in the semigroup tree approaches 0 as the genus grows to infinite. This means that most numerical semigroups have a finite number of descendants in the semigroup tree. This problem has been open since 2009.

math.NT↗
arXiv2023

On the seeds and the great-grandchildren of a numerical semigroup

We present a revisit of the seeds algorithm to explore the semigroup tree. First, an equivalent definition of seed is presented, which seems easier to manage. Second, we determine the seeds of semigroups with at most three left elements. And third, we find the great-grandchildren of any numerical semigroup in terms of its seeds. The RGD algorithm is the fastest known algorithm at the moment. But if one compares the originary seeds algorithm with the RGD algorithm, one observes that the seeds algorithm uses more elaborated mathematical tools while the RGD algorithm uses data structures that are better adapted to the final C implementations. For genera up to around one half of the maximum size of native integers, the newly defined seeds algorithm performs significantly better than the RGD algorithm. For future compilators allowing larger native sized integers this may constitute a powerful tool to explore the semigroup tree up to genera never explored before. The new seeds algorithm uses bitwise integer operations, the knowledge of the seeds of semigroups with at most three left elements and of the great-grandchildren of any numerical semigroup, apart from techniques such as parallelization and depth first search as wisely introduced in this context by Fromentin and Hivert. The algorithm has been used to prove that there are no Eliahou semigroups of genus $66$, hence proving the Wilf conjecture for genus up to $66$. We also found three Eliahou semigroups of genus $67$. One of these semigroups is neither of Eliahou-Fromentin type, nor of Delgado's type. However, it is a member of a new family suggested by Shalom Eliahou.

math.CO↗
arXiv2023

Isometry-Dual Flags of Many-Point AG Codes

Let $F_q$ be a finite field. A flag of $F_q$-linear codes $C_0\subsetneq C_1\subsetneq\dots\subsetneq C_s$ is said to satisfy the isometry-dual property if there exists a vector $x\in(F_q^*)^n$ such that $C_i=x\cdot C_{s-i}^\perp$, where $C_i^\perp$ denotes the dual code of $C_i$. Consider $F/F_q$ a function field and let $P$ and $Q_1,\ldots,Q_t$ be rational places of $F$. Let the divisor $D$ be the sum of pairwise different places of $F$ such that $P, Q_1,\dots,Q_t$ are not in $supp(D)$. In a previous work we investigated the existence of flags of two-point codes $C(D,a_0P+bQ_1)\subsetneq C(D,a_1P+bQ_1))\subsetneq\dots\subsetneq C(D,a_sP+bQ_1)$ satisfying the isometry-dual property for a non-negative integer $b$ and an increasing sequence of positive integers $a_0,\dots,a_s$. While for one-point codes (i.e. for $b=0$) there is only need to analyze positive integers $a$, for the case of $(t+1)$-point codes, the integers $a$ may be negative. We extend our previous results in different directions. On one hand to the case of negative integers $a$ and $b$, and on the other hand we extend our results to flags of $(t+1)$-point codes $C(D,a_0P+\sum_{i=1}^tβ_iQ_i)\subsetneq C(D, a_1P+\sum_{i=1}^tβ_iQ_i))\subsetneq\dots\subsetneq C(D, a_sP+\sum_{i=1}^tβ_iQ_i)$ for any tuple of (either positive or negative) integers $β_1,\dots,β_t$ and for an increasing sequence of (either positive or negative) integers $a_0,\dots,a_s$. We apply the obtained results to the broad class of Kummer extensions defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $gcd(r, m)=1$. In particular, depending on the place $P$ and for $D$ an $Aut(F_q(x, y)/F_q(x))$-invariant sum of rational places of $F$ such that $P,Q_i\notin supp(D)$, we obtain necessary and sufficient conditions on $m$ and $β_i$'s such that the flag has the isometry-dual property.

cs.IT↗
arXiv2021

The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes

A flag of codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}_q^n$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q^*)^n$ such that the code $C_i$ is {\bf x}-isometric to the dual code $C_{s-i}^\perp$ for all $i=0,\ldots, s$. For $P$ and $Q$ rational places in a function field ${\mathcal F}$, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes $$C_\mathcal L(D, a_0P+bQ)\subsetneq C_\mathcal L(D, a_1P+bQ)\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+bQ),$$ where the divisor $D$ is the sum of pairwise different rational places of ${\mathcal F}$ and $P, Q$ are not in $\mbox{supp}(D)$. We characterize those sequences in terms of $b$ for general function fields. We then apply the result to the broad class of Kummer extensions ${\mathcal F}$ defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $\mbox{gcd}(r, m)=1$. For $P$ the rational place at infinity and $Q$ the rational place associated to one of the roots of $f(x)$, it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if $m$ divides $2b+1$. At the end we illustrate our results by applying them to two-point codes over several well know function fields.

cs.IT↗
arXiv2021

The Key Equation for One-Point Codes

For Reed-Solomon codes, the key equation relates the syndrome polynomial---computed from the parity check matrix and the received vector---to two unknown polynomials, the locator and the evaluator. The roots of the locator polynomial identify the error positions. The evaluator polynomial, along with the derivative of the locator polynomial, gives the error values via the Forney formula. The Berlekamp-Massey algorithm efficiently computes the two unknown polynomials. This chapter shows how the key equation, the Berlekamp-Massey algorithm, the Forney formula, and another formula for error evaluation due to Horiguchi all generalize in a natural way to one-point codes. The algorithm presented here is based on Kötter's adaptation of Sakata's algorithm.

cs.IT↗
arXiv2021

Weierstrass semigroup at $m+1$ rational points in maximal curves which cannot be covered by the Hermitian curve

We determine the Weierstrass semigroup $H(P_\infty,P_1,\ldots,P_m)$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced by Tafazolian, Teherán-Herrera, and Torres. Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with better relative parameters than comparable one-point AG codes arising from these curves.

math.AG↗
arXiv2021

Quasi-ordinarization transform of a numerical semigroup

We introduce the quasi-ordinarization transform of a numerical semigroup. This transform will allow to organize all the semigroups of a given genus in a forest rooted at all quasi-ordinary semigroups with the given genus. This construction provides an alternative approach to the conjecture on the increasingness of the number of numerical semigroups for each given genus. We elaborate on the number of nodes at each tree depth in the forest and present a few new conjectures that can be developed in the future. We prove some properties of the quasi-ordinarization transform, its relations with the ordinarization transform, and we also present an alternative approach to the conjecture that the number of numerical semigroups of each given genus is increasing.

math.CO↗
arXiv2020

General Confidentiality and Utility Metrics for Privacy-Preserving Data Publishing Based on the Permutation Model

Anonymization for privacy-preserving data publishing, also known as statistical disclosure control (SDC), can be viewed under the lens of the permutation model. According to this model, any SDC method for individual data records is functionally equivalent to a permutation step plus a noise addition step, where the noise added is marginal, in the sense that it does not alter ranks. Here, we propose metrics to quantify the data confidentiality and utility achieved by SDC methods based on the permutation model. We distinguish two privacy notions: in our work, anonymity refers to subjects and hence mainly to protection against record re-identification, whereas confidentiality refers to the protection afforded to attribute values against attribute disclosure. Thus, our confidentiality metrics are useful even if using a privacy model ensuring an anonymity level ex ante. The utility metric is a general-purpose metric that can be conveniently traded off against the confidentiality metrics, because all of them are bounded between 0 and 1. As an application, we compare the utility-confidentiality trade-offs achieved by several anonymization approaches, including privacy models (k-anonymity and $ε$-differential privacy) as well as SDC methods (additive noise, multiplicative noise and synthetic data) used without privacy models.

cs.CR↗
arXiv2019

Patterns on Numerical Semigroups

We introduce the notion of pattern for numerical semigroups, which allows us to generalize the definition of Arf numerical semigroups. In this way infinitely many other classes of numerical semigroups are defined giving a classification of the whole set of numerical semigroups. In particular, all semigroups can be arranged in an infinite non-stabilizing ascending chain whose first step consists just of the trivial semigroup and whose second step is the well known class of Arf semigroups. We escribe a procedure to compute the closure of a numerical semigroup with respect to a pattern. By using the concept of system of generators associated to a pattern we construct recursively a directed acyclic graph with all the semigroups admitting the pattern.

math.RA↗
arXiv2019

The right-generators descendant of a numerical semigroup

For a numerical semigroup, we encode the set of primitive elements that are larger than its Frobenius number and show how to produce in a fast way the corresponding sets for its children in the semigroup tree. This allows us to present an efficient algorithm for exploring the tree up to a given genus. The algorithm exploits the second nonzero element of a numerical semigroup and the particular pseudo-ordinary case in which this element is the conductor.

math.CO↗