Any conditions on non-smooth functions you might know of?

Okay. The part that I was lacking is that, since $k$ points will define a hyperplane (in the large space of coefficients) uniquely: then the $k+1$ point can't be in a determined hyperplane. This happens with probability $1$. Thanks.

If I understand correctly, you say that a set of points are in general position, say with respect to degree $d$ hypersurfaces, if the linear equations they impose are independent. Not any linear equation would arise this way in the space of coefficients. In other words, given a probability measure over the set of points, you get a probability measure over the linear equations on the coefficient space that is supported on a small set. There is not much to say about this, but I would like to have some sort of reference (place where such a notion is defined, or at least used).

Thanks, That's pretty much the direction I am aiming at. 2 question: 1) Must I define general position with respect to the type of subojects or are there definitions in the literature, independent on the the type of subobjects, that work well on many interesting cases? 2) Is there a discussion in the literature that "justifies" such definitions of general position (in the sense that it captures the idea of randomly picked points) .

The circle was not a good example, I'll change it. Eric, can you state your notations? ($H^0$, $\mathcal{O}$) I am not too familiar with algebraic geometric notions)