Search

Given coprime positive integers $$g_1< \ldots < g_e$$ g 1 < … < g e , the Frobenius number $$F=F(g_1,\ldots ,g_e)$$ F = F ( g 1 , … , g e ) is the largest integer not representable as a linear combination of $$g_1,\ldots ,g_e$$ g 1 , … , g e with non-negative integer coefficients. Let n denote the number of all representable non-negative integers less than F; Wilf conjectured that $$F+1 \le e n$$ F + 1 ≤ e n . We provide bounds for $$g_1$$ g 1 and for the type of the numerical semigroup $$S=\langle g_1,\ldots ,g_e \rangle $$ S = ⟨ g 1 , … , g e ⟩ in function of e and n, and use these bounds to prove that $$F+1 \le q e n$$ F + 1 ≤ q e n , where $$q= \Bigg \lceil \frac{F+1}{g_1} \Bigg \rceil $$ q = ⌈ F + 1 g 1 ⌉ , and $$F+1 \le e n^2$$ F + 1 ≤ e n 2 . Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup $$S=\langle g_1,\ldots ,g_e \rangle $$ S = ⟨ g 1 , … , g e ⟩ is almost-symmetric.

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., Comment: 9 pages

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 $${\mathbb {A}}^4$$ A 4 is at most 3, answering a question of Stamate in this particular case. Moreover, we prove that, if $${\mathcal {C}}$$ C is a nearly Gorenstein affine monomial curve that is not Gorenstein and $$n_1, \dots , n_{\nu }$$ n 1 , ⋯ , n ν are the minimal generators of the associated numerical semigroup, the elements of $$\{n_1, \dots , \widehat{n_i}, \dots , n_{\nu }\}$$ { n 1 , ⋯ , n i ^ , ⋯ , n ν } are relatively coprime for every i.

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these resutls, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these results, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.

We prove that the Cohen–Macaulay type of an almost Gorenstein monomial curve C ⊆ A 4 is at most 3, and make some considerations on the general case.

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., Minor corrections

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 $\frac{S}{d}$., To appear in Journal of Commutative Algebra

Given coprime positive integers $$a_1 < \cdots < a_d$$ , 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 $$d \ge \frac{F +1}{F+1-g}$$ . 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 $$d = \frac{F+1}{F+1-g}$$ .

Given three pairwise coprime positive integers $a_1,a_2,a_3 \in \mathbb{Z}^+$ we show the existence of a relation between the sets of the first elements of the three quotients $\frac{\langle a_i,a_j \rangle}{a_k}$ that can be made for every $\{i.j,k\}=\{1,2,3\}$. Then we use this result to give an improved version of Johnson's semi-explicit formula for the Frobenius number $g(a_1,a_2,a_3)$ without restriction on the choice of $a_1,a_2,a_3$ and to give an explicit formula for a particular class of numerical semigroups., 7 pages