Grace–Walsh–Szegő theorem

Short description: Mathematical theorem about polynomials

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, ..., 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 ${\zeta }_1,\dots ,{\zeta }_n\in A$ there exists $\zeta \in A$ such that

$f({\zeta }_1,\dots ,{\zeta }_n)=f(\zeta ,\dots ,\zeta ).$