Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties

Abstract

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.

Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety

“…, γn) ∈ R n could be infinite or finite but can not be attained, i.e. c * 0 is an asymptotic critical value at infinity [21,22,26]. Hence, the proof of [30,Theorem 5.23] is not valid in this case.…”

Brouwer's degree without properness

“…The issue of the size of A(f ) has already been considered by algebraic geometers for different purposes: The smallness of A(f ) when f is a polynomial map was established by Jelonek in [5] (complex case) and [6] (real case). Generalizations to smooth algebraic varieties or to C 1 semialgebraic f can be found in Jelonek and Kurdyka [7] and Kurdyka, Orro and Simon [8], respectively. In these works, the algebraic structure makes it possible to express the smallness of A(f ) in terms of dimension.…”

On asymptotic critical values and the Rabier Theorem

“…As a consequence we give a direct proof of the fact that B(f ) ⊂ K(f ) in the case when X ⊂ k n is a smooth submanifold and f : X → k m is a smooth mapping (moreover, some of these results are used in [3] to study the properties of the set K(f )).…”

“…The case m > 1 and X = k n was studied in [1], [2] and [4]. In this paper (and in [3]) we study the case when X is a smooth affine variety (or even a Stein submanifold of C m ) and m ≤ dim X.…”

“…It can be proved that the set K(f ) is a proper algebraic subset of C m -or proper semi-algebraic in the real case (see [3]). Moreover, we have (e.g., by a general Fibration Theorem of Rabier [6], see also [1]) B(f ) ⊂ K(f ).…”