Bifurcation values and monodromy of mixed polynomials

Abstract

Abstract. We study the bifurcation values of real polynomial maps f : R 2n → R 2 , which reflect the lack of asymptotic regularity at infinity. We formulate real counterparts of some structure results, which have been previously proved in case of complex polynomials by Kushnirenko, Némethi and Zaharia and other authors, emphasizing the typical real phenomena that occur.

Invertible polynomial mappings via Newton non-degeneracy

“…-Let J F denote the set of points at which F is not proper (see One of the new issues of our paper is the non-degeneracy condition at infinity which appears to be a generic condition (Definition 3.4). This extends to mappings the definitions of "Newton non-degeneracy at infinity" for functions, both in the complex setting [18], [3], [4], [20] and in the more recently developed mixed setting [6]. Moreover, this works over the reals too.…”

“…In particular, in case k = 1 we get N (f ) = {0} for any non-convenient polynomial f . Let us remark that the set {0} appears as a component in the union of sets which occur as bound for the bifurcation set of a polynomial map B(f ) in the formula by Némethi-Zaharia [20] and also in the one by Chen-Tibăr [6].…”

Milnor fibration at infinity for mixed polynomials

“…For mixed polynomials, one can also ask under which condition does the Milnor fibration /| | at infinity exist? In this paper, we present an approach to this problem by using the strong non-degeneracy condition at infinity defined in [6]. Consider a mixed polynomial : C → C. Inspired by Oka's construction in the local case, we prove a similar result in the global setting.…”

“…Multiplying (4) by , we obtain A = λB (6) which implies AA = BB since λ ∈ S 1 . From (5), (6) and (a a) = 0, we therefore get AB = 0.…”

“…In this section, we review the definitions of Newton polyhedron and non-degeneracy conditions introduced in [6]. Let be a mixed polynomial.…”