Euclid Meets Bezout: Intersecting Algebraic Plane Curves with the Euclidean Algorithm

摘要

We can be quite sure that Euclid (ca. 325 to ca. 265 BC) and Etienne Bezout (1730-1783) never met. But we show here how Euclid's algorithm for polynomials can be used to find, with their multiplicities, the points of intersection of two algebraic plane curves. As a consequence, we obtain a simple proof of Bezout's Theorem, giving the total number of such intersections.