Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f=(f₁,...,f_m):X\to k^m$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function...

For an analytic function f:ℝⁿ,0 → ℝ,0 having a critical point at the origin, we describe the topological properties of the partition of the family of trajectories of the gradient equation ẋ = ∇f(x) attracted by the origin, given by characteristic exponents and asymptotic critical values.

We derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularities ${\Sigma }^{i,j}$. The formulas are given as linear combinations of Schur polynomials, and all coefficients are nonnegative.