On the enhancement to the Milnor number of a class of mixed polynomials. (English) Zbl 1296.14028
By definition, the enhancement to the Milnor number is an invariant of the homotopy classes of fibred links in the sphere \(S^{2n-1}\) containing either in \(\mathbb Z\) if \(n=2\) or in \(\mathbb Z/2\,\mathbb Z\) if \(n>2\) (see [W. D. Neumann and L. Rudolph, Lect. Notes Math. 1350, 109–121 (1988; Zbl 0655.57015)]). The author proves that any element of \(\mathbb Z\) and \(\mathbb Z/2\,\mathbb Z\) is realized by the enhancement to the Milnor number of the corresponding fibred link associated with a certain class of mixed polynomials, that is, convergent power series with complex coefficients of the form \(f(z,\bar z) = \sum c_{I,J}z^I{\bar z}^J,\) where \(z=(z_1,\dots, z_n),\) \(\bar z\) is the complex conjugate of \(z,\) and \(I=(i_1,\dots, i_n),\) \(J=(j_1,\dots, j_n)\) for \(n\geq 2.\)
Reviewer: Aleksandr G. Aleksandrov (Moskva)
MSC:
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 37C27 | Periodic orbits of vector fields and flows |
| 58K45 | Singularities of vector fields, topological aspects |
Keywords:
enhanced Milnor number; fibered link, open book decomposition; mixed polar weighted homogeneous polynomials; radial Newton polygon; Newton boundaryCitations:
Zbl 0655.57015References:
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