form (a, b, 0). Equivalently, (a, b) is a zero of pd. The fundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers), pd factors into a product of linear factors. Each factor defines a point at infinity on the curve: if bx − ay is such a factor, then it defines the point at infinity (a, b, 0). Over the reals, pd factors into linear and quadratic factors. The irreducible quadratic factors define non-real points at infinity, and the real points are given by the linear factors. If (a, b, 0) is a point at infinity of the curve, one says that (a, b) is an asymptotic direction. Setting q = pd the equation of the corresponding asymptote is If and the asymptote is the line at infinity, and, in the real case, the curve has a branch that looks like a parabola. In this case one says that the curve has a parabolic branch. If the curve has a singular point at infinity and may have several asymptotes. They may be computed by the method of computing the tangent cone of a singular point. Singular points The singular points of a curve of degree d defined by a polynomial p(x,y) of degree d are the solutions of the system of equations: In characteristic zero, this system is equivalent to where, with the notation of the preceding section, The systems are equivalent because of Euler's homogeneous function theorem. The latter system has the advantage of having its third polynomial of degree d-1 instead of d. Similarly, for a projective curve defined by a homogeneous polynomial P(x,y,z) of degree d, the singular points have the solutions of the system as homogeneous coordinates. (In positive characteristic, the equation has to be added to the system.) This implies that the number of singular points is finite as long as p(x,y) or P(x,y,z) is square free. Bézout's theorem implies thus that the number of singular points is at most (d − 1)2, but this bound is not sharp because the system of equations is overdetermined. If reducible polynomials are allowed, the sharp bound is d(d − 1)/2, this value is reached when the polynomial factors in linear factors, that is if the curve is the union of d lines. For irreducible curves and polynomials, the number of singular points is at most (d − 1)(d −