The differential equation satisfied by a plane curve of degree n

Eliminating the arbitrary coefficients in the equation of a generic plane curve of order n by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to conics, has been obtained by Monge. Sylvester, Halphen, Cartan used invariants of higher order. The expression of these invariants is rather complicated, but becomes much simpler when interpreted in terms of symmetric functions.

RésuméL'expression différentielle des courbes planes de degré donné fournit un invariant projectif. Monge a obtenu celle des coniques planes, Sylvester et Halphen ont généralisé l'équation de Monge aux courbes planes de tout degré. Nous montrons que la théorie des fonctions symétriques permet de retrouver ces invariants, et d'en donner des expressions plus compactes.