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Given a closed curve $C$ in the affine plane, we define its winding number around a point (that does not meet $C$) as the total number of times $C$ travels counter-clockwise around the point.

winding_-2 -2 winding_-1 -1 winding_0 0 winding_1 1 winding_2 2 winding_2 3

[images are from Wikipedia]

Suppose now that we are working over an algebraically closed field $k$ of characteristic zero. Do we have a similar notion for curves defined as the vanishing locus of a polynomial (that is, $C=V(f)$ with $f\in k[x,y]$ irreducible)?

Of course, it is not even clear to me how to tell whether $f$ defines a closed curve, and we have no orientation at all. Nevertheless, for regular curves, at least we have a plausible parametrisation, and for (the more interesting) singular ones, we could use blow-ups along the singular points.

As an example:

Consider the unit circle in $\mathbb{A}^2_{\mathbb{C}}$ given by $C=V(x^2+y^2-1)$. Then, for points inside the circle, the winding number should be one (we only care about the modulus), outside it should be zero.

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Hmmm. What applications of winding number are you hoping to capture here? Perhaps thinking about those will lead to an idea for a good definition, in the usual French style of turning theorems into definitions... –  Zhen Lin Dec 10 '12 at 0:52
    
Obviously, this has something to do with parametrisation of algebraic curves, which is related to the resolution of singular points (for a picture, think about a node). Hopefully, the winding number might control the number of subsequent blow-ups in a certain desingularisation process, or at least give us a better insight to the nature of singular curves. –  Norbert Pintye Dec 10 '12 at 1:57
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In your example, $\mathbb{A}^2\backslash C$ is connected (with the complex topology on $\mathbb{A}^2$), so how do we tell if a point is inside or outside $C$? –  Julian Rosen Dec 10 '12 at 2:14
    
You are right. The picture belongs to the real plane, but we need to work over an algebraically closed field. I have to think about this problem. –  Norbert Pintye Dec 10 '12 at 2:33
    
The standard algebro-geometric way of dealing with the question of fundamental group is the etale fundamental group. Perhaps you can approach algebro-geometric winding numbers with a similar philosophy. –  Aaron Mazel-Gee Dec 11 '12 at 0:34

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