# Algebraic vs. Analytic curves

I'm familiar with the idea of using algebra to study certain types of plane curves, and my understanding is that there is a whole class of "algebraic curves" that can be studied this way. It would also make sense that there are other plane curves, such as $y - \sin(x)=0$, that are not algebraic curves. Are there nevertheless algebra-like ways of studying these curves, and if not, why?

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Are you trying to refer to the "curves" studied in algebraic geometry? (Algebraic geometry refers to something very specific ; if you don't know what I'm talking about just say no, but I wanted to ask.) –  Patrick Da Silva Aug 16 '12 at 7:40
I guess I'm trying to avoid referring to "algebraic geometry" because then the answer is just "your curve is not a polynomial". I guess I could rephrase the question as "What makes polynomials special?", or "Are there ways to represent transcendental curves as polnomial-like structures that makes them ammenable to the tools os algebraic geometry?" but I'm starting to use words outside of my comfort zone at that point. –  Greg L Aug 16 '12 at 12:16
I don't know much about the words I'm about to use, but maybe that since $\sin(x)$ is analytic you could try using algebraic geometry on the limit of the curve with finitely many terms in the Taylor series. But I don't even know if anyone tried that before. –  Patrick Da Silva Aug 16 '12 at 12:53
@PatrickDaSilva, I was thinking along the same lines, that there'd be some sort of analytic framework to take the limit of some polynomial-like object, just not sure if this has been done or what it might be called. –  Greg L Aug 16 '12 at 22:50