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, 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 any $\zeta_1,\ldots,\zeta_n\in A$ 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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