00-371 David Ruelle

Grace-like polynomials (38K, plain tex) Sep 20, 00

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Abstract. Results of somewhat mysterious nature are known on the location of zeros of certain polynomials associated with statistical mechanics (Lee-Yang circle theorem) and also with graph counting. In an attempt at clarifying the situation we introduce and discuss here a natural class of polynomials. Let $P(z_1,\ldots,z_m,w_1,\ldots,w_n)$ be separately of degree 1 in each of its $m+n$ arguments. We say that $P$ is a Grace-like polynomial if $P(z_1,\ldots,w_n)\ne0$ whenever there is a circle in ${\bf C}$ separating $z_1,\ldots,z_m$ from $w_1,\ldots,w_n$. A number of properties and characterizations of these polynomials are obtained.

Files: 00-371.src( 00-371.keywords , grace.tex )