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Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by counting dimensions, the set $Z=\{(s,t): p(s,t)=0\}\subset \mathbb T^2$ is discrete.

Now assume that the original polynomial is perturbed slightly, $Q(z,w)=P(z,w)+R(z,w)$ where coefficients of $R(z,w)$ are small (in a controllable manner) w.r.t those of $P$.

Can I in principle estimate the magnitude of the corresponding perturbation of the set Z? (In the one-dimensional case, without the restriction on the circle, this would be just Rouche's theorem.)

I would like something like this: if the norm of the coefficients of $R$ is at most $\epsilon\ll 1$, then the distances of the elements of the new zero set $Z'=\{(s,t): Q(e^{is},e^{it})=0\}$ from their counterparts in $Z$ are bounded by some function of $\epsilon$.

I'm thinking about passing from $p$ to a meromorphic function by defining a suitable complex structure on $\mathbb T^2$ and then applying Rouche's theorem, but I don't quite understand how to do this properly.

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