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Algebraic curves

The asymptotes of an algebraic curve in the affine plane are the lines that are tangent to the projectivized curve through a point at infinity. For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves, although they also make sense when defined in this way for curves over an arbitrary field.

A plane curve of degree n intersects its asymptote at most at n − 2 other points, by Bézout's theorem, as the intersection at infinity is of multiplicity at least two. For a conic, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.

A plane algebraic curve is defined by an equation of the form P (x, y) = 0 where P is a polynomial of degree n

where P is homogeneous of degree k. Vanishing of the linear factors of the highest degree term P defines the asymptotes of the curve: setting Q = P, if P (x, y) = (ax − by) Q (x, y), then the line

is an asymptote if Q x ′ (b, a) {\displaystyle Q'_{x}(b,a)} and Q y ′ (b, a) {\displaystyle Q'_{y}(b,a)} are not both zero. If Q x ′ (b, a) = Q y ′ (b, a) = 0 {\displaystyle Q'_{x}(b,a)=Q'_{y}(b,a)=0} and P n − 1 (b, a) ≠ 0 {\displaystyle P_{n-1}(b,a)\neq 0}, there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called a parabolic branch, even when it does not have any parabola that is a curvilinear asymptote. If Q x ′ (b, a) = Q y ′ (b, a) = P n − 1 (b, a) = 0, {\displaystyle Q'_{x}(b,a)=Q'_{y}(b,a)=P_{n-1}(b,a)=0,} the curve has a singular point at infinity which may have several asymptotes or parabolic branches.

Over the complex numbers, P splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals, P splits in factors that are linear or quadratic factors. Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches. It may also occur that such a multiple linear factor corresponds to two complex conjugate branches, and does not corresponds to any infinite branch of the real curve. For example, the curve x + y - 1 = 0 has no real points outside the square | x | ≤ 1, | y | ≤ 1 {\displaystyle |x|\leq 1,|y|\leq 1}, but its highest order term gives the linear factor x with multiplicity 4, leading to the unique asymptote x =0.