Grace–Walsh–Szegő theorem

In mathematics, the Grace–Walsh–Szegő coincidence theorem[1][2] 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 $\zeta _{1},\ldots ,\zeta _{n}\in A$ there exists $\zeta \in A$ such that

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

Notes and references

"A converse to the Grace–Walsh–Szegő theorem", Mathematical Proceedings of the Cambridge Philosophical Society, August 2009, 147(02):447–453. doi:10.1017/S0305004109002424
J. H. Grace, "The zeros of a polynomial", Proceedings of the Cambridge Philosophical Society 11 (1902), 352–357.

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gollark: No.

gollark: The screen merely displays a pattern of light giving the impression of bees.

gollark: They aren't Obama any more than a computer screen displaying bees is bees.

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[1] "A converse to the Grace–Walsh–Szegő theorem", Mathematical Proceedings of the Cambridge Philosophical Society, August 2009, 147(02):447–453. doi:10.1017/S0305004109002424

[2] J. H. Grace, "The zeros of a polynomial", Proceedings of the Cambridge Philosophical Society 11 (1902), 352–357.