Grace–Walsh–Szegő theorem - Alchetron, the free social encyclopedia

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 ζ 1 , … , ζ n ∈ A there exists ζ ∈ A such that

f ( ζ 1 , … , ζ n ) = f ( ζ , … , ζ ) .

References

Grace–Walsh–Szegő theorem Wikipedia

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