On extremal quasiconformal extensions of univalent polynomials

Univalent polynomials form an important class of conformal maps and have deep applications in approximate construction of conformal maps and other fields.

The paper reveals rather surprising features of univalent polynomials whose finite critical points are placed on the boundary of the univalence disk. This provides explicitly the extremal quasiconformal dilatation and other quasiinvariants associated to such maps. The general case admits only much weaker estimates.