Related papers: On Euler-homogeneity for free divisors
In 2002, it was conjectured that a free divisor satisfying the so-called Logarithmic Comparison Theorem (LCT) must be strongly Euler-homogeneous. Today, it is known to be true only in ambient dimension less or equal than three or assuming…
We first develop some criteria for a general divisor to be strongly Euler-homogeneous in terms of the Fitting ideals of certain modules. We also study new variants of Saito-holonomicity, generalizing Koszul-free type properties and…
In this article, we prove that a free divisor in a three dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. F.J.…
We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing…
We introduce the concept of a prehomogeneous determinant as a nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the $b$-function are symmetric about -1 for…
Let D be a divisor in a complex analytic manifold X. A natural problem is to determine when the de Rham complex of meromorphic forms on X with poles along D is quasi-isomorphic to its subcomplex of logarithmic forms. In this mostly…
In this paper we study the comparison between the logarithmic and the meromorphic de Rham complexes along a divisor in a complex manifold. We focus on the case of free divisors, starting with the case of locally quasihomogeneous divisors,…
One deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both…
For a rank 1 local system on the complement of a reduced divisor on a complex manifold $X$, its cohomology is calculated by the twisted meromorphic de Rham complex. Assuming the divisor is everywhere positively weighted homogeneous, we…
Let $X$ be a smooth projective complex algebraic variety. An old question of Borel and Haefliger asks whether any (possibly singular) algebraic subvariety of $X$ is homologically equivalent to a linear combination with integral coefficients…
Given a linear elliptic equation $\sum a_{ij} u_{ij} =0$ in $\mathbb{R}^3$, it is a classical problem to determine if its degree-one homogeneous solutions $u$ are linear. The answer is negative in general, by a construction of…
By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation…
Master character of the multidimensional homogeneous Euler equation is discussed. It is shown that under restrictions to the lower dimensions certain subclasses of its solutions provide us with the solutions of various hydrodynamic type…
We investigate deformations of free and linear free divisors. We introduce a complex similar to the de Rham complex whose cohomology calculates deformation spaces. This cohomology turns out to be zero for many linear free divisors and to be…
This is a survey and research note on the modified Orlik conjecture derived from the division theorem introduced in [2]. The division theorem is a generalization of classical addition-deletion theorems for free arrangements. The division…
J.C.Lagarias (2000) conjectured that if $\mu$ is a complex measure on p-dimensional Euclidean space with a uniformly discrete support and its spectrum (Fourier transform) is also a measure with a uniformly discrete support, then the support…
In his paper and thesis in 1989, Ziegler posed several conjectures regarding commutative algebra related to hyperplane arrangements. In this article, we revisit two of them. One is on generic cuts of free arrangements, and the other has to…
We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring N_*(X) of a topological space X. This homology theory Eh_* has coefficients Z/2 in every nonnegative…
Recently A.Galajinsky has suggested the N=1 supersymmetric extension of Euler top and made a few interesting observations on its properties [arXiv:2111.06083 [hep-th]]. In this paper we use the formulation of the Euler top as a system on…
The equations for a self-similar solution of an inviscid incompressible fluid are mapped into an integral equation which hopefully can be solved by iteration. It is argued that the exponent of the similarity are ruled by Kelvin's theorem of…