Editing Bézout's theorem (section)

==Examples (plane curves)==
===Two lines===
The equation of a [[line (geometry)|line]] in a [[Euclidean plane]] is [[linear equation|linear]], that is, it equates a [[polynomial]] of degree one to zero. So, the Bézout bound for two lines is {{math|1}}, meaning that two lines either intersect at a single point, or do not intersect. In the latter case, the lines are [[parallel lines|parallel]] and meet at a [[point at infinity]].

One can verify this with equations. The equation of a first line can be written in [[slope-intercept form]] <math>y=sx+m</math> or, in [[projective coordinates]] <math>y=sx+mt</math> (if the line is vertical, one may exchange {{mvar|x}} and {{mvar|y}}). If the equation of a second line is (in projective coordinates) <math>ax+by+ct=0,</math> by substituting <math>sx+mt</math> for {{mvar|y}} in it, one gets <math>(a+bs)x + (c+bm)t=0.</math> If <math>a+bs\ne 0, </math> one gets the {{mvar|x}}-coordinate of the intersection point by solving the latter equation in {{mvar|x}} and putting {{math|1=''t'' = 1.}}

If <math>a+bs= 0, </math> that is <math>s=-a/b,</math> the two line are parallel as having the same slope. If <math>m\ne -c/b,</math> they are distinct, and the substituted equation gives {{math|1=''t'' = 0}}. This gives the point at infinity of projective coordinates {{math|(1, ''s'', 0)}}.

===A line and a curve===
As above, one may write the equation of the line in projective coordinates as <math>y=sx+mt.</math> If curve is defined in projective coordinates by a [[homogeneous polynomial]] <math>p(x,y,t)</math> of degree {{mvar|n}}, the substitution of {{mvar|y}} provides a homogeneous polynomial of degree {{mvar|n}} in {{mvar|x}} and {{mvar|t}}. The [[fundamental theorem of algebra]] implies that it can be factored in linear factors. Each factor gives the ratio of the {{mvar|x}} and {{mvar|t}} coordinates of an intersection point, and the multiplicity of the factor is the multiplicity of the intersection point.

If {{mvar|t}} is viewed as the ''coordinate of infinity'', a factor equal to {{mvar|t}} represents an intersection point at infinity.

If at least one partial derivative of the polynomial {{mvar|p}} is not zero at an intersection point, then the tangent of the curve at this point is defined (see {{slink|Algebraic curve|Tangent at a point}}), and the intersection multiplicity is greater than one if and only if the line is tangent to the curve. If all partial derivatives are zero, the intersection point is a [[Algebraic curve#Singular points|singular point]], and the intersection multiplicity is at least two.

===Two conic sections===
Two [[conic section]]s generally intersect in four points, some of which may coincide. To properly account for all intersection points, it may be necessary to allow complex coordinates and include the points on the infinite line in the projective plane. For example:
*Two circles never intersect in more than two points in the plane, while Bézout's theorem predicts four.  The discrepancy comes from the fact that every circle passes through the same two complex points on the line at infinity. Writing the circle <math display="block">(x-a)^2+(y-b)^2 = r^2</math> in [[homogeneous coordinates]], we get <math display="block">(x-az)^2+(y-bz)^2 - r^2z^2 = 0,</math> from which it is clear that the two points {{math|(1 : ''i'' : 0)}} and {{math|(1 : –''i'' : 0)}} lie on every circle.  When two circles do not meet at all in the real plane, the two other intersections have non-real coordinates, or if the circles are concentric then they meet at exactly the two points on the line at infinity with an intersection multiplicity of two.
*Any conic should meet the line at infinity at two points according to the theorem. A hyperbola meets it at two real points corresponding to the two directions of the asymptotes.  An ellipse meets it at two complex points, which are conjugate to one another{{mdash}}in the case of a circle, the points {{math|(1 : ''i'' : 0)}} and {{math|(1 : –''i'' : 0)}}.  A parabola meets it at only one point, but it is a point of tangency and therefore counts twice.
*The following pictures show examples in which the circle {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> – 1 = 0}} meets another ellipse in fewer intersection points because at least one of them has multiplicity greater than one:
{{Gallery|title=Intersection of an [[ellipse]] and the [[unit circle]]|
File:Bezout theorem1.svg|Two intersections of multiplicity 2<br/><math> x^2+4y^2-1=0</math>|
File:Bezout theorem2.svg|Two intersections of multiplicities 3 and 1<br/><math>5x^2+6xy+5y^2+6y-5=0</math>|
File:Bezout theorem3.svg|One intersection of multiplicity 4<br/><math>4x^2+y^2+6x+2=0</math>}}