 00371 David Ruelle
 Gracelike polynomials
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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
(LeeYang 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 Gracelike 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.
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