Desingularisation of the triple points and of the stationary points of a map
In a paper written in 1876 [4], Felix Klein gave a formula relating the number of real flexes of a generic real plane projective curve to the number of real bitangents at non-real points and the degree, which shows in particular that the number of real flexes cannot exceed one third of the total number of flexes. We show that Klein's arguments can be made rigorous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples.
À l’aide du Nullstellensatz effectif, on trouve des bornes inférieure et supérieure explicites des valeurs critiques non nulles d’un polynôme, en termes des coefficients de celui-ci.
It is a classical result, first established by de Jonquières (1859), that generically the number of conics tangent to 5 given conics in the complex projective plane is 3264. We show here the existence of configurations of 5 real conics such that the number of real conics tangent to them is 3264.
Nous classifions les transformations birationnelles quadratiques de l'espace projectif complexe de dimension trois, à des isomorphismes linéaires près. Elles sont de trois sortes, selon que le degré de leur inverse est 2, 3 ou 4. Il y a en tout 30 types différents; en 1871, L. Cremona en avait déjà décrit 23.
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