Families of polynomials with total Milnor number constant.
Remark on the equisingularity of families of affine plane curves
We give some criteria for the equisingularity of families of affine plane curves.
Topology of families of affine plane curves
We determine bifurcation sets of families of affine curves and study the topology of such families.
Sur le nombre de Łojasiewicz à l'infini d'un polynôme
Résumé. Soit f un polynôme à deux indéterminées. On appelle nombre de Łojasiewicz à l'infini de f le nombre de Łojasiewicz à l'infini de son application gradient. Dans cet article nous montrons tout d'abord que l'on peut calculer le nombre de Łojasiewicz d'un polynôme à partir des diagrammes de Eisenbud et Neumann de toutes les courbes f(x,y) = t. Ensuite nous montrons que l'on peut définir un nombre de Łojasiewicz intrinsèque en prenant le maximum des nombres de Łojasiewicz de f ∘ ϕ si f est bon...
Théorème de Kuiper—Kuo—Bochnak—Lojasiewicz à l'infini
On the Łojasiewicz exponent at infinity of real polynomials
Let f: ℝⁿ → ℝ be a nonconstant polynomial function. Using the information from the "curve of tangency" of f, we provide a method to determine the Łojasiewicz exponent at infinity of f. As a corollary, we give a computational criterion to decide if the Łojasiewicz exponent at infinity is finite or not. Then we obtain a formula to calculate the set of points at which the polynomial f is not proper. Moreover, a relation between the Łojasiewicz exponent at infinity of f and the problem of computing...
On the Łojasiewicz exponent near the fibre of polynomial mappings
We give the formula expressing the Łojasiewicz exponent near the fibre of polynomial mappings in two variables in terms of the Puiseux expansions at infinity of the fibre.
Łojasiewicz exponent of the gradient near the fiber
It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber such that r is the Łojasiewicz exponent of grad(f) near the fiber . We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula...
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