Grace–Walsh–Szegő theorem

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

Statement

Suppose ƒ(z₁, ..., z_n) is a polynomial with complex coefficients, and that it is

• symmetric, i.e. invariant under permutations of the variables, and
• multi-affine, i.e. affine in each variable separately.

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

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