Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.

We construct a family of high order iteration functions for finding polynomial roots of a known multiplicity s. This family is a generalization of a fundamental family of high order algorithms for ...

The main result of this paper is that, for any sequence s of positive integers, the s-Eulerian polynomial has only real roots. This result is first shown to generalize several existing results about ...