Bezout's Theorem

In general, two algebraic curves of degrees $m$ and $n$ intersect in $m\cdot n$ points and cannot meet in more than $m\cdot n$ points unless they have a component in common (i.e., the equations defining them have a common factor). This can also be stated: if $P$ and $Q$ are two Polynomials with no roots in common, then there exist two other Polynomials $A$ and $B$ such that $AP+BQ=1$. Similarly, given $N$ Polynomial equations of degrees $n_1$, $n_2$, ...$n_N$ in $N$ variables, there are in general $n_1n_2\cdots n_N$ common solutions.

See also Polynomial

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 10, 1959.