I was in a PhD crisis, but I did not post it anywhere. Not sure if it is allowed to ask for outside help. Although now I have finished the thesis without that

I was in a PhD crisis, but I did not post it anywhere. Not sure if it is allowed to ask for outside help. Although now I have finished the thesis without that part (it should have become an additional chapter). Perhaps I should post it around (although that might spoil it for a paper) Consider n polynomial equations in variables x_1, .., x_n, with constants a_1,..,a_n, b_1,..,b_n, c_1,...,c_n: p_1 := a_1 x_1 x_2 + b_1 x_1 + c_1 x_2 + d_1 = 0 ... p_{n-1} := a_{n-1} x_{n-1} x_n + b_{n-1} x_{n-1} + c_{n-1} x_n + d_n = 0 p_n := a_{n} x_{n} x_1 + b_{n} x_{n} + c_{n} x_1 + d_n = 0 Under which circumstances exists a (unique) solution for x_1,..,x_n in terms of the constants? I have found a recursive approach that results in a quadratic equation, containing only a single variable x_i (and the constants). (It is too much for a comment, here is a PDF: http://benibela.de/tmp/quadratic-equations-recursion.pdf http://benibela.de/tmp/quadratic-equations-recursion.pdf ) For example for n = 2, it is very simple: x^2_1 (a_2 b_1 - a_1 c_2) + x_1 (a_2 d_1 + b_2 b_1 - a_1 d_2 - c_1 c_2) + b_2 d_1 - c_1 d_2 This gives 2 solution. But I do not know what happens if the terms cancel each other out. Like if a_2 b_1 - a_1 c_2 = 0, there would only be one solution. But since the full solution in the pdf is so complex, I do not see which constraints would lead to cancellation there. --- And that is not the full problem I was trying to solve. In the full problem there are constraints on the a, b, c, d. There is a given graph, and depending which nodes are connected in the graph, the constants are the same. Like if node 3 and node 7 are connected, then b_3 = c_7 und c_7 = b_3. (even more complex though). And then the question is, do these constants cancel in the solution of those equations? And the final problem we want to solve: which graphs lead to exactly one solution, and which graphs lead to no solution of the equations?