Structurally stable perturbations of polynomials in the Riemann sphere
Sur les propriétés topologiques des projections lagrangiennes en géométrie symplectique des caustiques.
La caustique d?un point sur une variété riemannienne est l?ensemble des points d?intersection des géodésiques infiniment voisins partant de ce point. Jacobi a remarqué, en utilisant un raisonnement topologique, que la caustique d?un point sur une surface convexe fermée doit avoir des points de rebroussement. Il a aussi annoncé (sans démonstration) que le nombre de ces points est quatre pour les caustiques sur les surfaces d?ellipsoïdes (Jacobi, 1964). Dans cette note j?essaie d?inclure les théorèmes...
The Fedoryuk Condition and the Łojasiewicz Exponent near a Fibre of a Polynomial
We give a description of the set of points for which the Fedoryuk condition fails in terms of the Łojasiewicz exponent at infinity near a fibre of a polynomial.
The Łojasiewicz gradient inequality in a neighbourhood of the fibre
Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.
The Morse-Sard-Brown Theorem for Functionals on Bounded Fréchet-Finsler Manifolds
In this paper we study Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if is a connected smooth bounded Fréchet-Finsler manifold endowed with a connection and if is a smooth Lipschitz-Fredholm vector field on with respect to which satisfies condition (WCV), then, for any smooth functional on which is associated to , the set of the critical values of is of first category in . Therefore,...
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Valeurs critiques asymptotiques d'une fonction définissable dans une structure o-minimale
We prove that the set of asymptotic critical values of a function definable in an o-minimal structure is finite, even if the structure is not polynomially bounded. As a consequence, the function is a locally trivial fibration over the complement of this set.