Le's vanishing polyhedron for a family of mixed functions
Por:
Cisneros-Molina J.L., Menegon, Aurelio
Publicada:
1 dic 2019
Ahead of Print:
1 oct 2019
Categoría:
Mathematics (miscellaneous)
Resumen:
We study real analytic isolated singularities of type
f:(Cn+m+1,0)->(C,0) with f(z,w)=g(z)+ n-ary sumation i=1m+1hi(wi,w over
bar i), where g is holomorphic and each hi is a mixed polynomial, with
z=(z1,MIDLINE HORIZONTAL ELLIPSIS,zn) and w=(w1,MIDLINE HORIZONTAL
ELLIPSIS,wm+1). We construct a Le's vanishing polyhedron for f, which
describes the degeneration of its Milnor fiber Ff to the singular fiber.
Then we prove that Ff is homotopy equivalent to the join
Fg*Fh1*MIDLINE HORIZONTAL ELLIPSIS*Fhm+1, where Fg is the Milnor
fiber of g and Fhi is the Milnor fiber of hi. This implies that Ff has
the homotopy type of a bouquet of spheres Sn+m. So we can define the
Milnor number mu(f) as the number of spheres in that bouquet, as in the
complex setting.
Filiaciones:
Cisneros-Molina J.L.:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Avenida Universidad s/n, Chamilpa, Cuernavaca, Morelos, Mexico
Menegon, Aurelio:
Departamento de Matemática, Universidade Federal da Paraíba, CEP 58051–900 - Cidade Universitária, João Pessoa, Paraíba, Brazil
Univ Fed Paraiba, Dept Matemat, BR-58051900 Joao Pessoa, Paraiba, Brazil
Univ Nacl Autonoma Mexico, Inst Matemat, Ave Univ S-N, Cuernavaca, Morelos, Mexico
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