 096 David Ruelle
 Characterization of LeeYang polynomials
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Jan 7, 09

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Abstract. The LeeYang circle theorem describes complex polynomials of degree $n$ in $z$ with all their zeros on the unit circle $z=1$. These polynomials are obtained by taking $z_1=\ldots=z_n=z$ in certain multiaffine polynomials $\Psi(z_1,\ldots,z_n)$ which we call LeeYang polynomials (they do not vanish when $z_1,\ldots,z_n<1$ or $z_1,\ldots,z_n>1$). We characterize the LeeYang polynomials $\Psi$ in $n+1$ variables in terms of polynomials $\Phi$ in $n$ variables (those such that $\Phi(z_1,\ldots,z_n)\ne0$ when $z_1,\ldots,z_n<1$). This characterization gives us a good understanding of LeeYang polynomials and allows us to exhibit some new examples. In the physical situation where the $\Psi$ are temperature dependent partition functions, we find that those $\Psi$ which are LeeYang polynomials for all temperatures are precisely the polynomials with pair interactions originally considered by Lee and Yang.
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