Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry.

Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry.

Issu d un cours de maitrise de l Universite Paris VII, ce texte est reedite tel qu il etait paru en 1978. A propos du theoreme de Bezout sont introduits divers outils necessaires au developpement de la notion de multiplicite d intersection de deux courbes algebriques dans le plan projectif complexe. Partant des notions elementaires sur les sous-ensembles algebriques affines et projectifs, on definit les multiplicites d intersection et interprete leur somme entermes du resultant de deux polynomes. L etude locale est pretexte a l introduction des anneaux de serie formelles ou convergentes; elle culmine dans le theoreme de Puiseux dont la convergence est ramenee par des eclatements a celle du theoreme des fonctions implicites. Diverses figures eclairent le texte: on y "voit" en particulier que l equation homogene x3+y3+z3 = 0 definit un tore dans le plan projectif complexe.