Grace–Walsh–Szegő theorem

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
  1. ^ "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
  2. ^ J. H. Grace, "The zeros of a polynomial", Proceedings of the Cambridge Philosophical Society 11 (1902), 352–357.