Bézout's theorem

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Bézout's theorem

Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves "X" and "Y" is equal to the product of their degrees. This statement must be qualified in several important ways, by considering points at infinity, allowing complex coordinates (or more generally, coordinates from the algebraic closure of the ground field), assigning an appropriate multiplicity to each intersection point, and excluding a degenerate case when "X" and "Y" have a common component. A simpler special case is that if "X" and "Y" are both real or complex irreducible curves, "X" has degree "m" and "Y" has degree "n" then the number of intersection points does not exceed "mn".

Rigorous statement

Suppose that "X" and "Y" are two plane projective curves defined over a field "F" that do not have a common component (this condition is true if both "X" and "Y" are defined by different irreducible polynomials, in particular, it holds for a pair of "generic" curves). Then the total number of intersection points of "X" and "Y" with coordinates in an algebraically closed field "E" containing "F", counted with their multiplicities, is equal to the product of the degrees of "X" and "Y".

History

Some special cases of the theorem were known since the seventeenth century, especially in relation with intersections of lines, conics, and plane cubics. The theorem was published in 1776 in Étienne Bézout's "Théorie générale des équations algébriques". Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions. From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold. This led to a sentiment, expressed by certain authors, that his proof was neither correct nor the first proof to be given. [cite book | authorlink=Frances Kirwan | last=Kirwan | first=Frances | title=Complex Algebraic Curves | publisher=Cambridge University Press| location=United Kingdom | year=1992 | isbn=0-521-42353-8]

Intersection multiplicity

The most delicate part of Bézout's theorem and its generalization to the case of "k" algebraic hypersurfaces in "k"-dimensional projective space is the procedure of assigning the proper intersection multiplicities. If "P" is a common point of two plane algebraic curves "X" and "Y" that is a non-singular point of both of them and, moreover, the tangent lines to "X" and "Y" at "P" are distinct then the intersection multiplicity is one. This corresponds to the case of "transversal intersection". If the curves "X" and "Y" have a common tangent at "P" then the multiplicity is at least two. See intersection number for the definition in general.

Examples

*Two distinct non-parallel lines always meet in exactly one point. Two parallel lines intersect at a unique point that lies at infinity. To see how this works algebraically, in projective space, the lines "x"+2"y"=3 and "x"+2"y"=5 are represented by the homogeneous equations "x"+2"y"-3"z"=0 and "x"+2"y"-5"z"=0. Solving, we get "x"= -2"y" and "z"=0, corresponding to the point (-2:1:0) in homogeneous coordinates. As the "z"-coordinate is 0, this point lies on the line at infinity.

*The special case where one of the curves is a line can be derived from the fundamental theorem of algebra. In this case the theorem states that an algebraic curve of degree "n" intersects a given line in "n" points, counting the multiplicities. For example, the parabola defined by "y - x"2 = 0 has degree 2; the line "y" − "ax" = 0 has degree 1, and they meet in exactly two points when "a" ≠ 0 and touch at the origin (intersect with multiplicity two) when "a" = 0.

* Two conic sections 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 ::$\left(x-a\right)^2+\left(y-b\right)^2 = r^2$:in homogeneous coordinates, we get::$\left(x-az\right)^2+\left(y-bz\right)^2 - r^2z^2 = 0,$:from which it is clear that the two points (1:"i":0) and (1:-"i":0) lie on every circle. When two circles don't meet at all in the real plane (for example because they are concentric) they meet at these two points on the line at infinity and two other complex points which do not lie at infinity.:*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---in the case of a circle, the points (1:"i":0) and (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 "x"2+"y"2-1=0 meets another ellipse in fewer intersection points because at least one of them has multiplicity greater than 1:

::$x^2+4y^2-1=0: hbox\left\{two intersections of multiplicity 2\right\}$

::$5x^2+6xy+5y^2+6y-5=0: hbox\left\{an intersection of multiplicity 3\right\}$

::$4x^2+y^2+6x+2=0: hbox\left\{an intersection of multiplicity 4\right\}$

* [http://www.mathpages.com/home/kmath544/kmath544.htm Bezout's Theorem] at MathPages

References

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