Editing Intersection number (section)

== Intersection multiplicities for plane curves ==

There is a unique function assigning to each triplet <math>(P,Q,p)</math> consisting of a pair of projective curves, <math>P</math> and <math>Q</math>, in <math>K[x,y]</math> and a point <math>p \in K^2</math>, a number <math>I_p(P,Q)</math> called the ''intersection multiplicity'' of <math>P</math> and <math>Q</math> at <math>p</math> that satisfies the following properties:

# <math>I_p(P,Q) = I_p(Q,P)</math>
# <math>I_p(P,Q) = \infty</math> if and only if <math>P</math> and <math>Q</math> have a common factor that is zero at <math>p</math>
# <math>I_p(P,Q) = 0</math> if and only if one of <math>P(p)</math> or <math>Q(p)</math> is non-zero (i.e. the point <math>p</math> is not in the intersection of the two curves)
# <math>I_p(x,y) = 1</math> where <math>p = (0,0)</math>
# <math>I_p(P,Q_1Q_2) = I_p(P,Q_1) + I_p(P,Q_2)</math>
# <math>I_p(P + QR,Q) = I_p(P,Q)</math> for any <math>R \in K[x,y]</math>

Although these properties completely characterize intersection multiplicity, in practice it is realised in several different ways.

One realization of intersection multiplicity is through the dimension of a certain quotient space of the [[power series]] ring <math>K[[x,y]]</math>. By making a change of variables if necessary, we may assume that <math>p = (0,0)</math>. Let <math>P(x,y)</math> and <math>Q(x,y)</math> be the polynomials defining the algebraic curves we are interested in. If the original equations are given in homogeneous form, these can be obtained by setting <math>z = 1</math>. Let <math>I = (P,Q)</math> denote the ideal of <math>K[[x,y]]</math> generated by <math>P</math> and <math>Q</math>. The intersection multiplicity is the dimension of <math>K[[x,y]]/I</math> as a vector space over <math>K</math>.

Another realization of intersection multiplicity comes from the [[resultant]] of the two polynomials <math>P</math> and <math>Q</math>. In coordinates where <math>p = (0,0)</math>, the curves have no other intersections with <math>y = 0</math>, and the [[degree of a polynomial|degree]] of <math>P</math> with respect to <math>x</math> is equal to the total degree of <math>P</math>, <math>I_p(P,Q)</math> can be defined as the highest power of <math>y</math> that divides the resultant of <math>P</math> and <math>Q</math> (with <math>P</math> and <math>Q</math> seen as polynomials over <math>K[x]</math>).

Intersection multiplicity can also be realised as the number of distinct intersections that exist if the curves are perturbed slightly. More specifically, if <math>P</math> and <math>Q</math> define curves which intersect only once in the [[closure (mathematics)|closure]] of an open set <math>U</math>, then for a dense set of <math>(\epsilon,\delta) \in K^2</math>, <math>P - \epsilon</math> and <math>Q - \delta</math> are smooth and intersect transversally (i.e. have different tangent lines) at exactly some number <math>n</math> points in <math>U</math>. We say then that <math>I_p(P,Q) = n</math>.

=== Example ===
Consider the intersection of the ''x''-axis with the parabola <math>y = x^2 </math> at the origin.

Writing <math>P = y, </math> <math>Q = y - x^2,\ </math>and  <math>p = (0,0)</math> we get

: <math>I_p(P,Q) = I_p(y,y - x^2) = I_p(y,x^2) = I_p(y,x) + I_p(y,x) = 1 + 1 = 2.\,</math>

Thus, the intersection multiplicity is two; it is an ordinary [[tangent|tangency]]. Similarly one can compute that the curves <math>y = x^m </math> and <math>y = x^n </math> with integers <math>m>n\geq 0 </math> intersect at the origin with multiplicity ''<math>n. </math>''