Characterization of Lee-Yang polynomials

Abstract

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 z1 =···=zn = z in certain multiaffine polynomials ψ(z1,...,zn) which we call Lee-Yang polynomials (they do not vanish when |z1|,...,|zn| < 1 or |z1|,...,|zn| > 1). We characterize the Lee-Yang polynomials ψ in n+1 variables in terms of polynomials Φ in n variables (those such that Φ(z1,...,zn) ≠ 0 when |z1|,...,|zn| < 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.