Grace–Walsh–Szegő theorem
# 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 $ƒ(z_1,$ ..., $z_n)$ 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

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