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Suppose that we have two varieties $V,W$ (affine or projective, arbitrary). They are two algebraic object and as usual, we can define the map $\varphi: V\longrightarrow W$. It's easy to think about it as $\varphi(P)=(\varphi_{1}(P),...,\varphi_{m}(P))$ for any point $P=(a_1,...,a_{n})$ of $V$.

Then if we want to study the map $\varphi$, we have to care about $\varphi_{i} : V\longrightarrow k$, where $k$ is the base field.

My question is, why does $\varphi_{i}$ need to be a polynomial map ? Why do people care about the regular property? What did lead the mathematician to the idea of regular function ?

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    Oh. I see. You should just think of "regular" as "locally polynomial." It is analogous to the case of a smooth manifold. You want the functions between smooth manifolds to be "smooth," but the only way to do this is to define locally smooth. It is completely equivalent to say a function $f:M\to N$ is smooth if and only if for every smooth $\phi: N\to \mathbb{R}$ the composition $\phi\circ f: M\to \mathbb{R}$ is smooth. If you approach smooth manifolds from this viewpoint it is a direct translation that the definition of regular function is what corresponds to locally polynomial.2012-09-11

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There are several reasons that you have to think about the regularity of map, and you has talked about it in your comment. I am not sure about how the idea of regularity of map was developed, but here is a hint : Think about the local properties of a point on a variety, and consider the tangent space(that is quite nature).