The classical Grace–Walsh–Szegő Coincidence Theorem, see Rahman and Schmeisser (Analytic theory of polynomials, Oxford University Press, 2002, p. 108), relates the solutions of a symmetric multi-affine polynomial \(P(z_1,\ldots , z_n)\) , with complex coefficients, to the zeros of the univariate polynomial \(p(z):=P(z,\ldots ,z)\) . It says that every circular domain containing all points in a solution of P contains at least one zero of p. That result, together with its equivalent Grace’s theorem, see Grace (Proc. Cambr. Philos. 11:352–357, 1902) or Rahman and Schmeisser (2002, p. 107), are among the most important fundamental results in the geometric theory of polynomials. The goal of this work is to search for structures in the algebraic variety given by the solutions of a multi-affine polynomial, symmetric or not. Since multi-affine polynomials are closely related to Möbius transformations, it is natural to look for patterns made up of circles. Given a solution \((u_1,\ldots ,u_n)\) of \(P(z_1,\ldots , z_n)=0\) , we define the notion of weakly invariant circles given that solution. We give necessary and sufficient conditions for weakly invariant circles to exist, given the solution \((u_1,\ldots ,u_n)\) , and describe how to find them. In the second half of the paper we specialize our findings to symmetric multi-affine polynomials. Surprisingly, under very mild conditions, if \(P(z_1,\ldots , z_n)\) has weakly invariant circles given the solution \((u_1,\ldots ,u_n)\) , then the points \(u_1,\ldots ,u_n\) lie necessarily on a circle.
