Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties

  • 175 Accesses

  • 12 Citations


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.

Author information

Correspondence to Zbigniew Jelonek or Krzysztof Kurdyka.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jelonek, Z., Kurdyka, K. Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties. Discrete Comput Geom 34, 659–678 (2005).

Download citation


  • Computational Mathematic
  • Real Case
  • Quantitative Generalize
  • Affine Variety
  • Algebraic Subset