Compositio Mathematica

Compositio Mathematica (1999), 119:2:157-167 Cambridge University Press
Copyright © 1999 Kluwer Academic Publishers

Fibers of Polynomial Mappings at Infinity and a Generalized Malgrange Condition

a1 Northeastern University. Department of Mathematics, MA 02115-5096, Boston, U.S.A. e-mail:

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gaffney t   [Google Scholar] 


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 C$^∞$-triviality of f.

Key Words: polynomial mappings; singularity at infinity; characteristic cycle; Malgrange‘s Condition..