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Register a new userAbout the computation of the signature of surface singularities z^N+g(x,y)=0
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In this article we describe our experiences with a parallel SINGULAR-implementation of the signature of a surface singularity defined by z^N+g(x,y)=0.
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This paper is devoted to study the Lie algebra of linear symmetries of a homogenous 2nd order ODE, by the method of Kushner, Lychagin and Robstov.
Let $f$ be an isolated singularity at the origin of $mathbb{C}^n$. One of many invariants that can be associated with $f$ is its {L}ojasiewicz exponent $mathcal{L}_0 (f)$, which measures, to some extent, the topology of $f$. We give, for generic surface singularities $f$, an effective formula for $mathcal{L}_0 (f)$ in terms of the Newton polyhedron of $f$. This is a realization of one of Arnolds postulates.
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z}, ngeq 2, xyz eq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Bakers theory and Bilu-Hanrot-Voutiers result on primitive divisors of Lucas numbers.
We explicitly describe infintesimal deformations of cyclic quotient singularities that satisfy one of the deformation conditions introduced by Wahl, Kollar-Shepherd-Barron and Viehweg. The conclusion is that in many cases these three notions are different from each other. In particular, we see that while the KSB and the Viehw
In [9], Migliore, Miro-Roig and Nagel, proved that if $R = mathbb{K}[x,y,z]$, where $mathbb{K}$ is a field of characteristic zero, and $I=(L_1^{a_1},dots,L_r^{a_4})$ is an ideal generated by powers of 4 general linear forms, then the multiplication by the square $L^2$ of a general linear form $L$ induces an homomorphism of maximal rank in any graded component of $R/I$. More recently, Migliore and Miro-Roig proved in [8] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjecture that the same holds for arbitrary powers. In this paper we will solve this conjecture and we will prove that if $I=(L_1^{a_1},dots,L_r^{a_r})$ is an ideal of $R$ generated by arbitrary powers of any set of general linear forms, then the multiplication by the square $L^2$ of a general linear form $L$ induces an homomorphism of maximal rank in any graded component of $R/I$.
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