I am trying to solve a set of system of symbolic non-linear equations:

`g1 = ptz + pz + 2 pty q0 q1 - 2 ptz q1^2 + 2 px q0 q2 - 2 pz q2^2 - 2 px q1 q3 - 2 pty q2 q3 - 2 ptz q3^2 - 2 pz q3^2 ; g2 = 2 (ptx q0 q1 + px q0 q1 + ptz q1 q2 - pz q1 q2 + ptz q0 q3 + pz q0 q3 - ptx q2 q3 + px q2 q3); g3 = ptx + px - 2 ptx q1^2 - 2 px q1^2 - 2 pz q0 q2 - 2 pty q1 q2 - 2 px q2^2 - 2 pty q0 q3 - 2 pz q1 q3 - 2 ptx q3^2 ; g4 = -2 pty q0 q2 - 2 py q0 q2 + 2 ptz q1 q2 - 2 pz q1 q2 - 2 ptz q0 q3 - 2 pz q0 q3 - 2 pty q1 q3 + 2 py q1 q3 ; g5 = ptz + pz - 2 py q0 q1 - 2 pz q1^2 - 2 ptx q0 q2 - 2 ptz q2^2 - 2 ptx q1 q3 - 2 py q2 q3 - 2 ptz q3^2 - 2 pz q3^2 ; g6 = -pty - py - 2 pz q0 q1 + 2 py q1^2 + 2 ptx q1 q2 + 2 pty q2^2 + 2 py q2^2 - 2 ptx q0 q3 + 2 pz q2 q3 + 2 pty q3^2 ; g7 = q0^2 + q1^2 + q2^2 + q3^2; NSolve[{g1 == 0, g2 == 0, g3 == 0, g4 == 0, g5 == 0, g6 == 0, g7 == 1}, {q0, q1, q2, q3}, Reals] `

Here all variables except q0, q1, q2 and q3 are considered fixed. The variables represent a unit quaternion. Testing for corner cases (by setting single element of quaternion to 0) reveals that these set of equations don’t have a solution, which is what I intend to prove. But the code takes too long to run. Any suggestions would be appreciated.

I could treat the elements of quaternion and the permutations of the elements as separate variable and solve the system as Linear Equations, which I did for the corner cases. But here I don’t have enough constraints (10 unknowns with 7 constraints) and hence can’t employ that method.