Bézout's theorem  From Wikipedia, the free encyclopedia Jump to:
navigation, search This article is about Bézout's theorem in
algebraic geometry. For Bézout's theorem in arithmetic, see
Bézout's identity. Not to be confused with Little 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 when one does not care
about multiplicities and X and Y are two algebraic curves in the
Euclidean plane whose implicit equations are