# Grace–Walsh–Szegő theorem

Short description: Mathematical theorem about polynomials

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, ..., 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 $\displaystyle{ \zeta_1,\ldots,\zeta_n\in A }$ 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.