Let $Q$ be a real bivariate polynomial such that $Q(x,\tan(x))\geq 0$ for any $x\not\in\{\pm\frac{\pi}{2}\}+(2\pi){\mathbb Z}$. Does it necessarily follow that $Q$ is nonnegative on the whole of ${\mathbb R}^2$ ?

It is known that there is no non-trivial algebraic identity relating $x$ to $\tan(x)$, and that makes me think that the answer is YES.

  • $\begingroup$ Would not $Q(x,y) = x^2+(y-1)^2-\epsilon$ be a counterexample? $\endgroup$ – user147263 Sep 11 '14 at 22:09

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