Courbes algébriques planes = Plane Algebraic Curves
Publication year: 2008
ISBN: 978-3-540-33708-9
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Resulting from a master's course at the University of Paris VII, this text is re-edited as it appeared in 1978. Various tools are introduced in connection with Bézout's theorem necessary for the development of the notion of the multiplicity of intersection of two algebraic curves in the complex projective plane. Starting from elementary notions on affine and projective algebraic subsets, we define the intersection multiplicities and interpret their sum in terms of the resultant of two polynomials. The local study is a pretext for the introduction of formal or convergent series rings; it culminates in Puiseux's theorem, the convergence of which is reduced by splits to that of the theorem of implicit functions. Various figures illuminate the text: we "see" in particular that the homogeneous equation x3 + y3 + z3 = 0 defines a torus in the complex projective plane.
Subject: Mathematics and Statistics, Algebraic Geometry, Courbes algébriques planes, Polygone de Newton, Singularités, Théorème de Puiseux, Théorème de Bézout, Plane algebraic curves, Newton's polygon, Singularities, Puiseux's theorem, Bézout's theorem
