Critical values of singularities at infinity of complex polynomials (original) (raw)

Bifurcation Sets and Global Monodromies of Newton Nondegenerate Polynomials on Algebraic Sets

Publications of the Research Institute for Mathematical Sciences

Let S ⊂ C n be a non-singular algebraic set and f : C n → C be a polynomial function. It is well-known that the restriction f | S : S → C of f on S is a locally trivial fibration outside a finite set B(f | S) ⊂ C. In this paper, we give an explicit description of a finite set T ∞ (f | S) ⊂ C such that B(f | S) ⊂ K 0 (f | S) ∪ T ∞ (f | S), where K 0 (f | S) denotes the set of critical values of the f | S. Furthermore, T ∞ (f | S) is contained in the set of critical values of certain polynomial functions provided that the f | S is Newton non-degenerate at infinity. Using these facts, we show that if {f t } t∈[0,1] is a family of polynomials such that the Newton polyhedron at infinity of f t is independent of t and the f t | S is Newton non-degenerate at infinity, then the global monodromies of the f t | S are all isomorphic.

Results on Milnor Fibrations for Mixed Polynomials with Non-isolated Singularities

Bulletin of the Brazilian Mathematical Society, New Series, 2020

In this article we investigate mixed polynomials and present conditions that can be applied on a specific class of polynomials in order to prove the existence of the Milnor Fibration, Milnor-L Fibration and the equivalence between them. We prove for this class the of functions that the Milnor-Lê fiber on a regular value is homeomorphic to the Milnor-Lê fiber on a critical value. We develop a criterion to verify the transversality property and apply it to a special case of the class of mixed polynomial.

Bifurcation set, M-tameness, asymptotic critical values and Newton polyhedrons

Kodai Mathematical Journal, 2013

Let F = (F 1 , F 2 ,. .. , Fm) : C n → C m be a polynomial dominant mapping with n > m. In this paper we give the relations between the bifurcation set of F and the set of values where F is not M-tame as well as the set of generalized critical values of F. We also construct explicitly a proper subset of C m in terms of the Newton polyhedrons of F 1 , F 2 ,. .. , Fm and show that it contains the bifurcation set of F. In the case m = n − 1 we show that F is a locally C ∞-trivial fibration if and only if it is a locally C 0-trivial fibration.

Fibers of Polynomial Mappings at Infinity and a Generalized Malgrange Condition

Compositio Mathematica - COMPOS MATH, 1999

Let f be a complex polynomial mapping. We relate the behaviour of f ‘at infinity’ to the characteristic cycle associated to the projective closures of fibres of f. We obtain a condition on the characteristic cycle which is equivalent to a condition on the asymptotic behaviour of some of the minors of the Jacobian matrix of f. This condition generalizes the condition in the hypersurface case known as Malgrange's condition. The relation between this condition and the behavior of the characteristic cycle is a partial generalization of Parusinski's result in the hypersurface case. We show that the new condition implies the C8-triviality of f.

Topology of polynomial functions and monodromy dynamics

Comptes Rendus De L Academie Des Sciences Serie 1 Mathematique, 1998

R&urn& Let f : C"-t 63 be a polynomial function. We define Slobal polar invariants associated to fibres of f and we describe a CW-complex model of a fibre. We show how to use affine polar curves in order to study the monodromy around atypical values of f, including the value infinity. We give a zeta-function formula for such a monodromy. 0 AcadCmie des SciencesElsevier, Paris Topologie des fonctions polynomiales et dynamique monodromique Soit f : C"-+ a3 une fonction polynomiale. On d&jinit des invariants polaires globaux assock% aux jibres de f ri 1 'aide desquels on dkcrit un modtYe d'une jibre de f comme CW-complexe. On montre comment utiliser les courbes polaires afines pour ktudier la monodromie autour d'une valeur atypique, y compris la valeur injnie. On donne une formule pour la fonction zeta d'une telle monodromie. 0 AcadCmie des SciencesMsevier, Paris Version fraryaise abrt?g&e On etudie une fonction polynomiale f : 43"-+ 43, n 2 2, ayant comme but de d&ire la variation dans la topologie de la fibre de f due a la presence des fibres atypiques. Une valeur to E 43 est typique pour f si l'application f est une C"-fibration triviale en to. L-ensemble des valeurs atypiques est fini (c$ [8], [12]) et il inclut l'ensemble des valeurs critiques de f. Les valeurs atypiques non critiques sont dues au comportement asymptotique (< mauvais B d'un certain nombre de fibres. On definit d'abord des invariants polaires globaux y* (introduits dans [ 1 l] dans le but de construire une thtorie d'equisingularitt5 globale des families d'hypersurfaces affines) et on montre comment ces y* entrent dans la description d'un modele CW-complexe d'une fibre de f (theoreme 1.3). Note prhsentke par Bernard MALGRANGE.