Complex System of Equations

Algebra Level 5

For how many ordered pairs \((a,b)\) of complex numbers does the system of equations \[ \begin{cases} x^2y-x=a\\ xy^2+y=b \end{cases}\] have exactly one complex number solution?

Details and assumptions

A complex number solution is an ordered pair of complex numbers \((x,y)\) that satisfies both equations.

Clarification: \(a, b, x\) and \(y\) are all complex numbers. As an explicit example, \( (x,y) = (i, i) \) is a solution to \((a,b) = (-2i, 0 ) \).


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