Defining G(E,F). Let E be a polynomial equation with n variables. Let E be at most quadratic in any variable. Let F be a field or a ring. Define a graph G(E,F) to hava a vertex set made up of n-tuples from Fn which are solutions to E. Let the edges in G be when two solutions differ in exaclty one variable.
Examples of multivariable quadratics:
2(a2 + b2 + c2 + d2) = (a + b + c + d)2 [Soddy]
a2 + b2 + c2 = 3 a b c [Markoff]
a12 + a22 + a32 + ... + an2 = B a1 a2 a3 ... an [Hurwitz]
a2 + b2 + 2 c2 = 4 a b c [Markoff Brother #1]
a2 + 2 b2 + 3 c2 = 6 a b c [Markoff Brother #2]
Note that The Soddy Equation was proved by Descartes.
If M is the markoff equation, then G(M,Z) has 5 components. Martin says that for all of the p's he has tried, G(M,Z/p) is connected.
G(M,GF(p^k)) does not seem to be connected for k > 1.