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).