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, ..., zn) 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 ζ 1 , … , ζ n ∈ A {\displaystyle \zeta _{1},\ldots ,\zeta _{n}\in A} there exists ζ ∈ A {\displaystyle \zeta \in A} such that
f ( ζ 1 , … , ζ n ) = f ( ζ , … , ζ ) . {\displaystyle f(\zeta _{1},\ldots ,\zeta _{n})=f(\zeta ,\ldots ,\zeta ).}
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