This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not even so much interested in characterizing the solutions as I am in counting them (or showing some relation between them), but any help would be greatly appreciated (as well as a pointer to a good introductory article/text on quadratic and modular forms in general).

Thanks!

nondegeneratequadratic form over F_p can be written in diagonal form as x_1^2 + ... + x_{n-1}^2 + dx_n^2, where the nonzero number d only matters modulo squares (so basically there are just two choices). For degenerate quadratic forms in n variables you just use fewer variables in the diagonalized form (like Q = x^2 as a function of x and y). $\endgroup$ – KConrad Aug 21 '10 at 5:17nota silly question, by the way. $\endgroup$ – Pete L. Clark Aug 21 '10 at 11:03