# Grace–Walsh–Szegő theorem

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In mathematics, the **Grace–Walsh–Szegő coincidence theorem**^{[1]}^{[2]} is a result named after John Hilton Grace, Joseph L. Walsh, 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 any simply connected open set in the complex plane. If either *A* is convex or the degree of *ƒ* is *n*, then for any there exists such that