Grace–Walsh–Szegő theorem

In mathematics, the Grace–Walsh–Szegő coincidence theorem is a result named after John Hilton Grace, Joseph L. Walsh, and Gábor Szegő.

Statement
Suppose ƒ(z1, ..., zn) is a polynomial with complex coefficients, and that it is Let A be a circular region in the complex plane. If either A is convex or the degree of ƒ is n, then for every $$\zeta_1,\ldots,\zeta_n\in A$$ there exists $$\zeta\in A$$ such that
 * symmetric, i.e. invariant under permutations of the variables, and
 * multi-affine, i.e. affine in each variable separately.


 * $$ f(\zeta_1,\ldots,\zeta_n) = f(\zeta,\ldots,\zeta). $$