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
"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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