For example, how to solve the equation $\sum^{p-1}\_{i}x\_{i}^{2}=0$ in $F\_{p}$? This is not a homework problem. I think it should have a definite answer, so not an open problem. I just don't know how to solve it. 



How to solve Diophantine equations in $F_{p}$?
There is a *deterministic* polynomial-time algorithm for finding solutions to diagonal equations of degree less than or equal to the number of variables over finite fields. See [Christiaan van de Woestijne's thesis](http://www.opt.math.tugraz.at/~cvdwoest/).


(A solution of your example equation can be found much more simply, however: try small integers, not necessarily distinct... . And for quadratic forms, the other solutions can be found by drawing lines through the point and intersecting with the quadric hypersurface: there will either be one more intersection point, or a whole line of points.)


