Complex multi-affine polynomials and invariant circles

Abstract

The classical Grace-Walsh-Szegő Coincidence Theorem relates the zeros of a polynomial p(z) with the points in the solution of the unique symmetric multi-affine polynomial P(z1,…,zn), such that p(z)=P(z,…,z). Thus, the geometric properties of the solutions of P(z1,…,zn) reveal some information of the zeros of p(z). Several geometric properties related to invariant circles of multi-affine polynomials were studied in [10] and [11]. Unfortunately, they seem to be too restrictive, when the degree of the polynomial is large, say four or more. This work investigates a relaxed notion for circles (C1,…,Cn) to be invariant with respect to multi-affine polynomials P of degree n, given a vector (u1,…,un). This weaker definition allows us to formulate necessary and sufficient conditions for invariant circles to exist, given (u1,…,un). In the case when the multi-affine polynomials P are symmetric, these necessary and sufficient conditions take a very simple and attractive form.