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Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it})$ its restriction to the torus. In the specific problem I'm considering, the set $Z=\{(s,t): p(s,t)=0\}\subset \mathbb T$ is discrete.

EDIT: $Z$ is discrete in the general position by counting dimensions, so I just assume that my specific situation is non-degenerate.

Now my 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? Meaning that, if the norm of the coefficients of $R$ is $\epsilon$, then the distance of the elements of the new zero set $Z'$ from the corresponding elements of $Z$ is some bounded function of $\epsilon$.

Note: in the one-dimensional case (without the restriction to the circle) this would be just Rouche's theorem.

  • 0
    OK, clarified this.2012-11-29

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