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 jzj D 1. These polynomials are obtained by taking z1 D D zn D z in certain multiaffine polynomials ‰.z1; : : : ; zn/ which we call Lee-Yang polynomials (they do not vanish when jz1j; : : : ; jznj 1). We characterize the Lee-Yang polynomials ‰ in nC1 variables in terms of polynomials ˆ in n variables (those such that ˆ.z1; : : : ; zn/¤ 0 when jz1j; : : : ; jznj 1 (including jzi j D 1 in a sense to be made precise later). Our current understanding of Lee-Yang polynomials is based on the concept of Asano contraction [1]. We shall define an inner radius associated with a multiaffine polynomial ˆ, and see that it behaves supermultiplicatively with respect to Asano contraction (Proposition 2). Using the properties of the inner radius, we shall characterize the ‰ 2 LYnC1 (nC 1 variables) in terms of polynomials ˆ in n variables such that ˆ.z1; : : : ; zn/¤ 0 when jz1j; : : : ; jznj < 1 (Theorem 3). This characterization will give us a good understanding of LYnC1 (Proposition 5), and allow us to exhibit elements of LYnC1 outside of the (pair interaction) class originally considered by Lee and Yang (see in particular Example 7(d)). The original