To show how powerful our new definitions are, we describe Bézout’s Theorem, which counts all the intersections of two algebraic curves keeping in mind their multiplicities. We sketch a proof of the theorem in the case when one of the two curves is a line. In this case, the theorem justifies our claim in the last paragraph of the previous chapter.
The details of the proof do not recur again in our journey, but their beauty justifies our small excursion.
Sometimes, we can’t get everything that we want, and sometimes we want something general rather than something specific. For example, ...