37,003 research outputs found
The L\^e numbers of the square of a function and their applications
L\^e numbers were introduced by Massey with the purpose of numerically
controlling the topological properties of families of non-isolated hypersurface
singularities and describing the topology associated with a function with
non-isolated singularities. They are a generalization of the Milnor number for
isolated hypersurface singularities.
In this note the authors investigate the composite of an arbitrary
square-free f and . They get a formula for the L\^e numbers of the
composite, and consider two applications of these numbers. The first
application is concerned with the extent to which the L\^e numbers are
invariant in a family of functions which satisfy some equisingularity
condition, the second is a quick proof of a new formula for the Euler
obstruction of a hypersurface singularity. Several examples are computed using
this formula including any X defined by a function which only has transverse
D(q,p) singularities off the origin.Comment: 14 page
Non-isolated Hypersurface Singularities and L\^e Cycles
In this series of lectures, I will discuss results for complex hypersurfaces
with non-isolated singularities. In Lecture 1, I will review basic definitions
and results on complex hypersurfaces, and then present classical material on
the Milnor fiber and fibration. In Lecture 2, I will present basic results from
Morse theory, and use them to prove some results about complex hypersurfaces,
including a proof of L\^e's attaching result for Milnor fibers of non-isolated
hypersurface singularities. This will include defining the relative polar
curve. Lecture 3 will begin with a discussion of intersection cycles for proper
intersections inside a complex manifold, and then move on to definitions and
basic results on L\^e cycles and L\^e numbers of non-isolated hypersurface
singularities. Lecture 4 will explain the topological importance of L\^e cycles
and numbers, and then I will explain, informally, the relationship between the
L\^e cycles and the complex of sheaves of vanishing cycles.Comment: Notes from a series of lectures from the S\~ao Carlos singularities
meeting of 2014. Revision made to Exercise 3.1 (a
Computation of Milnor numbers and critical values at infinity
We describe how to compute topological objects associated to a polynomial map
of several complex variables with isolated singularities. These objects are:
the affine critical values, the affine Milnor numbers for all irregular fibers,
the critical values at infinity, and the Milnor numbers at infinity for all
irregular fibers. Then for a family of polynomials we detect parameters where
the topology of the polynomials can change. Implementation and examples are
given with the computer algebra system Singular.Comment: 9 pages.To download the libraries for Singular see
http://www-gat.univ-lille1.fr/~bodin
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