2)/2, because of the formula expressing the genus in term of the singularities (see below). The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below). The equation of the tangents at a singular point is given by the nonzero homogeneous part of the lowest degree in the Taylor series of the polynomial at the singular point. When one changes the coordinates to put the singular point at the origin, the equation of the tangents at the singular point is thus the nonzero homogeneous part of the lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part. Analytic structure The study of the analytic structure of an algebraic curve in the neighborhood of a singular point provides accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point and look either as a cusp or as a smooth curve. Near a regular point, one of the coordinates of the curve may be expressed as an analytic function of the other coordinate. This is a corollary of the analytic implicit function theorem, and implies that the curve is smooth near the point. Near a singular point, the situation is more complicated and involves Puiseux series, which provide analytic parametric equations of the branches. For describing a singularity, it is worth to translate the curve for having the singularity at the origin. This consists of a change of variable of the form where are the coordinates of the singular point. In the following, the singular point under consideration is always supposed to be at the origin. The equation of an algebraic curve is where is a polynomial in and . This polynomial may be considered as a polynomial in , with coefficients in the algebraically closed field of the Puiseux series in . Thus may be factored in factors of the form where is a Puiseux series. These factors are all different if is an irreducible polynomial, because this implies that is square-free, a property which is independent of the field of coefficients. The Puiseux series that occur here have the form where is a positive integer, and is an integer that may also be supposed to be positive, because we