Grace–Walsh–Szegő theorem explained
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
there exists
such that
f(\zeta1,\ldots,\zetan)=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.
- J. H. Grace, "The zeros of a polynomial", Proceedings of the Cambridge Philosophical Society 11 (1902), 352–357.
