In mathematics, the Grace–Walsh–Szegő coincidence theorem 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
ζ
1
,
…
,
ζ
n
∈
A
{\displaystyle \zeta _{1},\ldots ,\zeta _{n}\in A}
there exists
ζ
∈
A
{\displaystyle \zeta \in A}
such that
f
(
ζ
1
,
…
,
ζ
n
)
=
f
(
ζ
,
…
,
ζ
)
.
{\displaystyle f(\zeta _{1},\ldots ,\zeta _{n})=f(\zeta ,\ldots ,\zeta ).}