Milnor fibration at infinity for mixed polynomials

Abstract

Abstract:We study the existence of Milnor fibration on a big enough sphere at infinity for a mixed polynomial :By using strongly non-degenerate condition, we prove a counterpart of Némethi and Zaharia's fibration theorem.In particular, we obtain a global version of Oka's fibration theorem for strongly non-degenerate and convenient mixed polynomials. MSC:14D06, 58K05, 57R45, 14P10, 32S20, 58K15

Real Polynomial Maps and Singular Open Books at Infinity

“…The condition p(t) ∈ M (f /|f |) is expressed by the following equation (see [Ch2] and [Ch1, Def. 5.2.2]):…”

“…One has to express df (p(t)) dt and like done in [Ch1] and [Ch2], take the real parts Re(·) and find that:…”

“…In [CT], [Ch1] and [Ch2] we have adapted the notion of Newton non-degeneracy to mixed polynomials and, more generally, to maps (real, mixed or complex) in [CDTT]. One has for instance the following result [Ch2,Theorem 3.5]: if f is Newton strongly non-degenerated on every face of the convex hull of its support, then M (f /|f |) is bounded. This is an extension of a result by for complex polynomials.…”

“…This is an extension of a result by for complex polynomials. Moreover, [Ch2,Theorem 3.6] shows that in order to prove condition (ii) it is enough to show that the critical loci of f △ |f △ | are empty on C * n for any strictly bad face △ of supp(f ), see the next example.…”

Singularities in Geometry and Topology

“…Other authors (for instance [27,8]) call polar weighted homogeneous functions to more general notions allowing the p i 's or q i 's to be zero; we call this more general definition generalized polar weighted homogeneous functions to emphasize the difference. Originally, the angular weights were called polar weights and this has caused some confusion in the literature because some authors (for instance [28,8]) call polar weighted homogeneous to mixed functions which are weighted homogeneous with respect to the angular weights and not to both radial and angular weights. To avoid this ambiguity in [5] the authors propose to use the term mixed weighted homogeneous instead of what we call polar weighted homogeneous.…”