Grace–Walsh–Szegő theorem

In mathematics, the Grace–Walsh–Szegő coincidence theorem{{cite journal

| last1=Grace | first1=J. H. | authorlink1=John Hilton Grace

| title=The zeros of a polynomial

| journal=Mathematical Proceedings of the Cambridge Philosophical Society

| volume=11

| date=1902

| pages=352–357

| url=https://archive.org/details/proceedingscamb13socigoog/page/352/mode/2up}}{{cite journal

| last1=Brändén | first1=Petter

| last2=Wagner | first2=David G.

| title=A converse to the Grace–Walsh–Szegő theorem

| journal=Mathematical Proceedings of the Cambridge Philosophical Society

| date=August 2009

| volume=147

| issue=2

| pages=447–453

| doi=10.1017/S0305004109002424| arxiv=0809.3225

}} 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

Category:Theorems in complex analysis

Category:Theorems about polynomials