Grace–Walsh–Szegő theorem
Appearance
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
[edit]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
Notes and references
[edit]- ^ Grace, J. H. (1902). "The zeros of a polynomial". Mathematical Proceedings of the Cambridge Philosophical Society. 11: 352–357.
- ^ Brändén, Petter; Wagner, David G. (August 2009). "A converse to the Grace–Walsh–Szegő theorem". Mathematical Proceedings of the Cambridge Philosophical Society. 147 (2): 447–453. arXiv:0809.3225. doi:10.1017/S0305004109002424.