Let $𝔞\subseteq ℂ[x_1,...,x_n]$ be a monomial ideal and $𝒥\left({𝔞}^c\right)$ the multiplier ideal of $𝔞$ with coefficient $c$. Then $𝒥\left({𝔞}^c\right)$ is also a monomial ideal of $ℂ[x_1,...,x_n]$, and the equality $𝒥\left({𝔞}^c\right)=𝔞$ implies that $0<c<n+1$. We mainly discuss the problem when $𝒥\left(𝔞\right)=𝔞$ or $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ for all $0<\epsilon <1$. It is proved that if $𝒥\left(𝔞\right)=𝔞$ then $𝔞$ is principal, and if $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ holds for all $0<\epsilon <1$ then $𝔞=(x_1,...,x_n)$. One global result is also obtained. Let $\widetilde{𝔞}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $𝔞$. Then it is proved that the equality $𝒥\left(\widetilde{𝔞}\right)=\widetilde{𝔞}$ implies that $\widetilde{𝔞}$ is principal.

On this paper we compute the numerical function $\beta$ of the approximation theorem of M. Artin for the one-dimensional systems of formal equations.

It is shown that every polynomial function P:ℂ² → ℂ with irreducible fibres of the same genus must be a coordinate. Consequently, there do not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves with the same genus.

Let $C$ be a curve of genus $g\ge 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $char\left(K\right)=0$ and that the characteristic $p$ of the residue field is not $2$. Suppose that the Jacobian $Jac\left(C\right)$ has semi-stable reduction over $R$. Embed $C$ in $Jac\left(C\right)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique tamely ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion...

In this short note we give an elementary combinatorial argument, showing that the conjecture of J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández and A. Némethi [Proc. London Math. Soc. 92 (2006), 99-138, Conjecture 1] follows from Theorem 5.4 of Brodzik and Livingston [arXiv:1304.1062] in the case of rational cuspidal curves with two critical points.

This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q,p,a}:y^q=x^p+a$, and its Jacobians $J_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3,5,a}\left(\mathbb{Q}\right)$ (resp. $J_{q,p,a}\left(\mathbb{Q}\right)$). The main tools are computations of the zeta function of $C_{3,5,a}$ (resp. $C_{q,p,a}$) over ${}_l$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp))...