Christian Hägg: Sequential differentiation of polynomials with zeros determined by simple polygons

Abstract: Consider a complex polynomial P with simple zeros in lattice points contained in a simple polygon S. We numerically investigate how the zeros of P, P', P'', ... change, and notice that they converge on trees. By instead considering a polynomial p with zeros of multiplicity n in the vertices of S, we see that the zeros of the n:th derivative of p reside in more refined trees or forests. These organic shapes seem to, given light restrictions, be contained in unique, simple polygons on the vertices of S.