In general, two algebraic curves of degrees and intersect in points and cannot meet in more than points unless they have a component in common (i.e., the equations defining them have a common factor). This can also be stated: if and are two Polynomials with no roots in common, then there exist two other Polynomials and such that . Similarly, given Polynomial equations of degrees , , ... in variables, there are in general common solutions.
See also Polynomial
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 10, 1959.