case. To compute the intersection of the curve defined by the polynomial p with the line of equation ax+by+c = 0, one solves the equation of the line for x (or for y if a = 0). Substituting the result in p, one gets a univariate equation q(y) = 0 (or q(x) = 0, if the equation of the line has been solved in y), each of whose roots is one coordinate of an intersection point. The other coordinate is deduced from the equation of the line. The multiplicity of an intersection point is the multiplicity of the corresponding root. There is an intersection point at infinity if the degree of q is lower than the degree of p; the multiplicity of such an intersection point at infinity is the difference of the degrees of p and q. Tangent at a point The tangent at a point (a, b) of the curve is the line of equation , like for every differentiable curve defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric: where is the derivative at infinity. The equivalence of the two equations results from Euler's homogeneous function theorem applied to P. If the tangent is not defined and the point is a singular point. This extends immediately to the projective case: The equation of the tangent of at the point of projective coordinates (a:b:c) of the projective curve of equation P(x, y, z) = 0 is and the points of the curves that are singular are the points such that (The condition P(a, b, c) = 0 is implied by these conditions, by Euler's homogeneous function theorem.) Asymptotes Every infinite branch of an algebraic curve corresponds to a point at infinity on the curve, that is a point of the projective completion of the curve that does not belong to its affine part. The corresponding asymptote is the tangent of the curve at that point. The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case. Let be the decomposition of the polynomial defining the curve into its homogeneous parts, where pi is the sum of the monomials of p of degree i. It follows that and A point at infinity of the curve is a zero of p of the