Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties

Abstract

Let X ⊂ ℂn be a smooth affine variety of dimension n – r and let f = (f1,..., fm): X → ℂm 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 ℂm. 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 fi ≤ d, then K(f) is contained in an algebraic hypersurface of degree at most (d + (m – 1)(d – 1)+(D – 1)r)n-rDr. 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-1D generalized 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.