Multivariable Quadratics Solution Graphs

Given a polynomial equation in 3 (for any value of 3) variables, which is at most quadratic in any variable, what does the solution space look like? See more information.

In a finite field of order p^k, the solutions to Markoff:

p = 2, k = 2, number solutions = 17,	number components = 3
p = 2, k = 3, number solutions = 65,	number components = 4
p = 2, k = 4, number solutions = 257,	number components = 5
p = 3, k = 2, number solutions = 81,	number components = 17
p = 3, k = 3, number solutions = 729,	number components = 92
p = 3, k = 4, number solutions = 6561,	number components = 881
p = 5, k = 2, number solutions = 701,	number components = 8
p = 5, k = 3, number solutions = 16001,	number components = 7

For k=1, these solutions seem to be connected (except for (0,0,0) solution, which is always a singleton). For k>1, this is not true. How many connected components?


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