| Speaker: Sione Ma'u Affiliation: Auckland Time: 2:00 pm Wednesday, 13 November, 2019 Location: 303-G14 |
| Let K be a compact body in (complex) n-dimensional space. Given a polynomial p such that the maximum value of |p| is 1 on K, how big can |p(z)| be at a point z outside of K? The best answer to this question is given by the (Siciak-Zaharjuta) extremal function of K. First I will recall the basic properties of this function. Then I will show how to compute it explicitly in certain cases. When K is a real simplex there is a formula for the extremal function in terms of barycentric coordinates, which can be deduced from work of M. Baran in the early 1990s. Recently (with Len Bos and Shayne Waldron) we gave a direct proof of the formula. Finally, I will show how to compute the extremal function of a real polytope using extremal functions for simplices. |