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arXiv 0811.1327math-phmath.MP
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Lee-Yang多项式的特征

Characterization of Lee-Yang polynomials

David Ruelle

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14 pages
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摘要：

李-杨圆定理描述了$n$次的复多项式，在$z$中，其所有零点都在单位圆$|z|=1$上。这些多项式是通过在某些多仿射多项式$Ψ(z_1,...,z_n)$中取$z_1=...=z_n=z$得到的，我们称之为李杨多项式（当$|z_1|,...,|z_n|<1$或$|z_1|,...,|z_n|>1$时它们不为零）。我们描述了李杨多项式$Ψ$在$n+1$变量的多项式$Φ$在$n$变量（那些使得$Φ(z_1,...,z_n)\ne0$时，$|z_1|,...,|z_n|<1$）。这一特征使我们对李杨多项式有了更好的理解，并使我们能够展示一些新的例子。在$Ψ$是温度相关配分函数的物理情形下，我们发现对所有温度都是李-杨多项式的那些$Ψ$正是李和杨最初考虑的具有对相互作用的多项式。

英文摘要：

The Lee-Yang 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 $z_1=...=z_n=z$ in certain multiaffine polynomials $Ψ(z_1,...,z_n)$ which we call Lee-Yang polynomials (they do not vanish when $|z_1|,...,|z_n|<1$ or $|z_1|,...,|z_n|>1$). We characterize the Lee-Yang polynomials $Ψ$ in $n+1$ variables in terms of polynomials $Φ$ in $n$ variables (those such that $Φ(z_1,...,z_n)\ne0$ when $|z_1|,...,|z_n|<1$). This characterization gives us a good understanding of Lee-Yang polynomials and allows us to exhibit some new examples. In the physical situation where the $Ψ$ are temperature dependent partition functions, we find that those $Ψ$ which are Lee-Yang polynomials for all temperatures are precisely the polynomials with pair interactions originally considered by Lee and Yang.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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