I am currently studying the chapter $3$ of Fulton's book on Algebraic Curves.The first section starts with the following comment:
We have seen that affine plane curves correspond to non-constant polynomials $F\in K[X,Y]$ without multiple factors,where $F$ is determined up to multiplication by a nonzero constant.For some purposes it is useful to allow $F$ to have multiple factors,so we modify our definition slightly:
We say that two polynomials $F,G\in K[X,Y]$ are equivalent if $F=\lambda G$ for some nonzero $\lambda\in K$.We define an affine plane curve to be an equivalence class of nonconstant polynomials under this equivalence relation.
Now this thing leads to a confusion in understanding.What does the author mean by multiple factors.Does it mean a scalar factor like $F=\lambda G$ where $\lambda\in K\setminus \{0\}$ or it means that $F(X,Y)=(X+Y)(X-Y)$ as well as $G(X,Y)=(X+Y)^2(X-Y)$ is allowed although they trace the same zero locus.
Can someone please clarify?
It is the latter, ie under this definition $(X + Y)(X - Y)$ and $(X+Y)^2(X - Y)$ both define (inequivalent) "affine curves". The issue of multiple factors is essentially unrelated to the issue of scalar multiples.