Characterization of Lee-Yang polynomials

The Lee-Yang circle theorem describes complex polynomials of degree n in z with all their zeros on the unit circle vertical bar z vertical bar = 1. These polynomials are obtained by taking z(1) = ... = z(n) = z in certain multiaffine polynomials Psi(z(1),...,z(n)) which we call Lee-Yang polynomials (they do not vanish when vertical bar z(1)vertical bar,..., vertical bar z(n)vertical bar < 1 or vertical bar z(1)vertical bar,...,vertical bar z(n)vertical bar > 1). We characterize the Lee-Yang polynomials Psi in n + 1 variables in terms of polynomials Phi in n variables (those such that Phi(z(1),...,z(n)) not equal 0 when vertical bar z(1)vertical bar,...,vertical bar z(n)vertical bar < 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 Psi are temperature dependent partition functions, we find that those Psi which are Lee-Yang polynomials for all temperatures are precisely the polynomials with pair interactions originally considered by Lee and Yang.