arXiv subjects
Publications and source records attributed to Alessio Moscariello.
We collect some open problems about minimal presentations of numerical semigroups and, more generally, about defining ideals and free resolutions of their semigroup rings and associated graded rings. We emphasize both long-standing problems and more recent questions and developments.
In this work we provide an upper bound for the multiplicity of a one-dimensional Cohen-Macaulay ring (under certain conditions), describe the rings attaining the equality for this bound, and outline a connection with Wilf's conjecture for numerical semigroup rings. Then we prove the analogue of Wilf's conjecture for almost Gorenstein rings.
In the last two decades there has been a wealth of results determining the precise value of the catenary degree and the tame degree. Mostly, however, only for very special classes of monoids and domains. In the present work we now show that there is no polynomial formula, neither for the catenary nor for the tame degree, which is valid for a sufficiently large class of finitely generated monoids.
We prove that the type of nearly Gorenstein numerical semigroups minimally generated by $5$ integers is bounded. In particular, if such a semigroup is not almost symmetric, then its type is at most $40$. Finally, we make some considerations in higher embedding dimension.
Let G be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of G which depends only on the width of G, that is, the difference between the largest and the smallest generator of G. In this way, we make progress towards a conjecture of Herzog and Stamate. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.
We consider the classical problem of determining the largest possible cardinality of a minimal presentation of a numerical monoid with given embedding dimension and multiplicity. Very few values of this cardinality are known. In addressing this problem, we apply tools from Hilbert functions and free resolutions of artinian standard graded algebras. This approach allows us to solve the problem in many cases and, at the same time, identify subtle difficulties in the remaining cases. As a by-product of our analysis, we deduce results for the corresponding problem for the type of a numerical monoid.
In this paper, by using a combinatorial approach, we establish a new upper bound for the F-threshold $c^\mm(\mm)$ of determinantal rings generated by maximal minors. We prove that $c^\mm(\mm)$ coincides with the $a$-invariant in the case of $3\times n$ and $4\times n$ matrices and we conjecture such equality holds for all matrices.
We prove that the Cohen-Macaulay type of an almost Gorenstein monomial curve $\mathcal{C} \subseteq \mathbb{A}^5$ is bounded.
Given coprime positive integers $g_1 < \ldots < g_e$, the Frobenius number $F=F(g_1,\ldots,g_e)$ is the largest integer not representable as a linear combination of $g_1,\ldots,g_e$ with non-negative integer coefficients. Let $n$ denote the number of all representable non-negative integers less than $F$; Wilf conjectured that $F+1 \le e n$. We provide bounds for $g_1$ and for the type of the numerical semigroup $S=\langle g_1,\ldots,g_e \rangle$ in function of $e$ and $n$, and use these bounds to prove that $F+1 \le q e n$, where $q= \left \lceil \frac{F+1}{g_1} \right \rceil$, and $F+1 \le e n^2$. Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup $S=\langle g_1,\ldots,g_e \rangle$ is almost-symmetric.
This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some applications to Universal Hilbert Sets generated by closed forms of linear recurrence relations and to integer perfect powers having few digits in their representation in a given scale $x \ge 2$ are provided.
This work is devoted to proving that, given an integer $x \ge 2$, there are infinitely many perfect powers, coprime with $x$, having exactly $k \ge 3$ non-zero digits in their base $x$ representation, except for the case $x=2, k=4$, for which a known finiteness result by Corvaja and Zannier holds.
Given three positive integers $a,b,c$, a proportionally modular Diophantine inequality is an expression of the form $ax \mod{b} \le cx$. Our aim is to give a recursive formula for the least solution to such an inequality. We then use the formula to derive an algorithm. Finally, we apply our results to a question of Rosales and García-Sánchez.
We extend some results on almost Gorenstein affine monomial curves to the nearly Gorenstein case. In particular, we prove that the Cohen-Macaulay type of a nearly Gorenstein monomial curve in $\mathbb{A}^4$ is at most $3$, answering a question of Stamate in this particular case. Moreover, we prove that, if $\mathcal C$ is a nearly Gorenstein affine monomial curve which is not Gorenstein and $n_1, \dots, n_ν$ are the minimal generators of the associated numerical semigroup, the elements of $\{n_1, \dots, \widehat{n_i}, \dots, n_ν\}$ are relatively coprime for every $i$.
This paper provides a formula for the minimal relations and the Frobenius number of a numerical semigroup minimally generated by three pairwise coprime positive integers.
We prove that the Cohen-Macaulay type of an almost Gorenstein monomial curve $\mathcal C \subseteq \mathbb{A}^4$ is at most $3$, and make some considerations on the general case.
Given a numerical semigroup $S$ and a positive integer $d$, the fraction $\frac{S}{d}=\{ x \in \mathbb{N} \ | \ dx \in S\}$ is again a numerical semigroup. In this paper we determine a generating set for $\frac{S}{d}$ in terms of the minimal generators of $S$ and provide sharp upper bounds for the embedding dimension of $\frac{S}{d}$.
Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive integers: Wilf conjectured that $d \geq \frac{F+1}{F+1-g}$. We prove that for every fixed value of $\lceil \frac{a_1}{d} \rceil$ the conjecture holds for all values of $a_1$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $d = \frac{F+1}{F+1-g}$.
We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set $\{n^2,(n+1)^2,\ldots \}$ is asymptotically $O(n^2)$, verifying thus a conjecture of Dutch and Rickett. Furthermore we ask a question on the representation of integers as sum of four large squares.