I am interested in whether there exist nontrivial (i.e., non-constant) solutions to any or all of the equations \begin{align*} P(x, x^{-1})^2 + Q(x, x^{-1})^2 + 1 &= 0, \\ P(x, x^{-1})^2 + Q(x, x^{-1})^4 + 1 &= 0, \\ P(x, x^{-1})^2 + Q(x, x^{-1})^3 + 1 &= 0, \\ P(x, x^{-1})^2 + Q(x, x^{-1})^3 + Q(x, x^{-1}) &= 0, \\ P(x, x^{-1})^2 + Q(x, x^{-1})^5 + 1 &= 0, \end{align*} where $P$ and $Q$ are single-variable Laurent polynomials with coefficients in $\mathbb{C}$. For example, a straightforward solution to the first equation is \begin{equation*} P(x, x^{-1}) = \frac{x - x^{-1}}{2}, \quad Q(x, x^{-1}) = \frac{i(x + x^{-1})}{2}. \end{equation*} I would like to know whether the other equations have nontrivial solutions, and if so, whether there exists a method for generating solutions or for classifying all solutions. Comments on the more general equation $P(x, x^{-1})^a + Q(x, x^{-1})^b + 1 = 0$, where $a$ and $b$ are positive integers, would be appreciated as well.

  • $\begingroup$ So we are not looking for simultaneous solutions, but for every equation separately? For example, the last equation has a solution $Q(x,x^{-1})=0$ and $P(x,x^{-1})=i$. $\endgroup$ – Dietrich Burde Nov 5 at 8:58
  • $\begingroup$ @Dietrich Burde Right, I should have been more clear - I am looking for solutions to each equation separately. Also, I am interested in solutions where $P$ and $Q$ are not constant. $\endgroup$ – user137 Nov 5 at 16:52
  • $\begingroup$ A version now posted to MO, mathoverflow.net/questions/345519/… $\endgroup$ – Gerry Myerson Nov 8 at 6:12

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