Results 151 to 160 of about 8,612,691 (178)

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

Functional Analysis and Its Applications, 2022
The Milnor number of an isolated complete intersection is greater than or equal to its Tjurina number if and only if the hypersurface is quasi-homogeneous. To calculate the Milnor and Tjurina numbers, the Poincaré-de Rham complex and the cotangent complex are useful.
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Milnor number of a hypersurface at the origin

2021
The modern theory of applications of Newton polyhedra to affine Bezout problem started from A. Kushnirenko’s work aimed at answering V. I. Arnold’s question on Milnor numbers of generic singularities. In [Kou76] Kushnirenko gave a beautiful formula for a lower bound of the Milnor number at the origin in terms of volumes of the region bounded by the ...
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Milnor numbers of nonisolated saito singularities

Functional Analysis and Its Applications, 1987
It is shown that Milnor numbers of a quasihomogeneous Saito singularity can be calculated by investigating the cohomology groups of a complex on certain affine space.
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Equivariant Milnor Numbers and Invariant Morse Approximations

Journal of the London Mathematical Society, 1985
Let G be a finite group, V an orthogonal complex representation of G and f: (V,0)\(\to {\mathbb{C}}\) the germ of a G-invariant holomorphic function with an isolated critical point. This paper proves that there is a deformation of a representative of f, through invariant functions, in which the generic fibre has only non-degenerate (or Morse) critical ...
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On Milnor's triple linking number

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1997
Summary: We define an operation of summation of 3 knots along a \(Y\)-graph, similar to the band sum of 2 components. Starting from the second degree Vassiliev knot invariant, we obtain, by means of \(Y\)-summation, Milnor's triple linking number \(\overline \mu_{123}\).
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A New Deterministic Method for Computing Milnor Number of an ICIS

2021
The Milnor number of an isolated complete intersection singularity (ICIS) is considered in the context of symbolic computation. Based on the classical Le-Greuel formula, a new method for computing Milnor numbers is introduced. Key ideas of our approach are the use of auxiliary indeterminates and the concept of local cohomology with coefficients in the ...
Shinichi Tajima, Katsusuke Nabeshima
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The Invariance of Milnor's Number Implies Topological Triviality

American Journal of Mathematics, 1977
THEOREM. Let F(z, t) be a polynomial in z = (z0, ... , zn) with coefficients which are smooth complex valued functions of t E RP such that F(O t) = 0, and for each t E RP, the polynomials aF/azi(z, t) in z have an isolated zero at 0. Assume moreover that the Milnor numbers ,t are independent of t, and that n # 2.
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Bounding Poincaré‐Hopf indices and Milnor numbers

Mathematische Nachrichten, 2005
AbstractWe use Mather's finite determinacy theory and Baum‐Bott's theorem to give sharp bounds for the Poincaré‐Hopf index of a germ of homolorphic vector field with an isolated zero. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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The enhanced Milnor number in higher dimensions

1988
The "enhanced Milnor number" of a fibered link was introduced homotopy theoretically in [N-R I] . We recall its definition later. It lies in Z(~Z or Z(~(Z/2) according as the ambient dimension is 3 or greater than 3. Its first component is, up to sign, the usual Milnor number, which is the dimension of the Seifert form if the fibered link is simple. We
Walter D. Neumann, Lee Rudolph
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A hyperplane section theorem for Milnor numbers

Mathematische Annalen, 1997
We prove the following result. Theorem. Let \(R\) denote the power series ring \(\mathbb{C} [[X_1,X_2, \dots, X_n]]\) and \(f\in R\) any irreducible element. Assume that for any element \(h\in R\) which is a part of a minimal system of generators of the maximal ideal of \(R\) the ring \(R/(f,h)\) has an isolated singular point.
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