If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma:=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when is $\Gamma$ contained in the closed unit disk?

Really this question is about applying a Theorem of Bereha, Kahane and Weiss which says that if $a_{1}(z)$, $\lambda_{1}(z)$, $a_{2}(z)$ and $\lambda_{2}(z)$ are all polynomials and

$P_{n}=a_{1}(z) \lambda_{1}(z)^{n}+a_{2}(z) \lambda_{2}(z)^{n}$

then the zeros of the $P_{n}(z)$ converge to some isolated points and the curve where $\lambda_{1}$ and $\lambda_{2}$ are equi-modular as $n\rightarrow \infty$. If $\lambda_{1}(z)=1$ and $\lambda_{2}(z)=z^{m}$ then the equimodular curve is the unit circle. Is there any other pair of polynomials whose equimodular curve is the unit circle? Or contained in the unit disk?

These equimodular curves often arise in the study of the zeros of the Tutte polynomial when it is considered as the Potts model partition function as they represent phase transitions.

I've read a bit on the topic (N. Biggs gave some methods for finding such curves) but am I am finding that my Google Scholar searches are becoming too broad!

Any suggestions? Right now I am just doing examples and trying to find out what's our there on the topic.

  • $\begingroup$ Interesting question! $\endgroup$ – Per Alexandersson Apr 17 '15 at 20:28

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