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, ..., 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 any simply connected open set in the complex plane. If either A is convex or the degree of ƒ is n, then for any ${\displaystyle \zeta _{1},\ldots ,\zeta _{n}\in A}$ there exists ${\displaystyle \zeta \in A}$ such that

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

Notes and references

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.