No abstract

No abstract

No abstract

In this paper we study the set of points at which a real polynomial mapping is not proper. (1991): 14 P, 26 C05. Mathematics Subject Classification

Let X ⊂ C n be a smooth affine variety of dimension n − r and let f = ( f 1 , . . . , f m ): X → C m be a polynomial dominant mapping. We prove that the set K ( f ) of generalized critical values of f (which always contains the bifurcation set B( f ) of f ) is a proper algebraic subset of C m . We give an explicit upper bound for the degree of a hypersurface containing K ( f ). If I (X )-the ideal of X -is generated by polynomials of degree at most D and deg f i ≤ d, then K ( f ) is contained in an algebraic hypersurface of degree at most (d + (m − 1)(d − 1) + (D − 1)r ) n−r D r . In particular if X is a hypersurface of degree D and f : X → C is a polynomial of degree d, then f has at most (d + D − 1) n−1 Dgeneralized critical values. This bound is asymptotically optimal for f linear. We give an algorithm to compute the set K ( f ) effectively. Moreover, we obtain similar results in the real case.

No abstract

No abstract

No abstract