Grace–Walsh–Szegő theorem

In mathematics, the Grace–Walsh–Szegő coincidence theorem^[1][2] is a result named after John Hilton Grace, Joseph L. Walsh, and 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 ${\displaystyle \zeta _{1},\ldots ,\zeta _{n}\in A}$ there exists ${\displaystyle \zeta \in A}$ such that

${\displaystyle f(\zeta _{1},\ldots ,\zeta _{n})=f(\zeta ,\ldots ,\zeta ).}$