Grace–Walsh–Szegő theorem
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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 [math]\displaystyle{ \zeta_1,\ldots,\zeta_n\in A }[/math] there exists [math]\displaystyle{ \zeta\in A }[/math] such that
- [math]\displaystyle{ f(\zeta_1,\ldots,\zeta_n) = f(\zeta,\ldots,\zeta). }[/math]
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.
Original source: https://en.wikipedia.org/wiki/Grace–Walsh–Szegő theorem.
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