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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

Very much depends on the level. Many basic properties of curves are studied using the differential and integral calculus. If you are using techniques from the calculus, it does not matter very much whether the curve is algebraic or transcendental.

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Sorry, what do you mean by "depends on the level"? If you are implying that my example is too simple to merit a more robust algebraic approach, that is not quite what I'm after. I'd like to know if such an approach exists regardless. I'm specifically looking for the existence of techniques beyond "the calculus". – Greg L Aug 16 '12 at 22:53
@Greg L.: You will find a rich vein of algebraic studies connected with your question by looking under differential algebra. – André Nicolas Aug 21 '12 at 18:17
Thanks for the reference! – Greg L Aug 25 '12 at 20:52

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