Uniform triangulations and generalized derangement polynomials

• Applicant: Professorin Dr. Martina Juhnke-Kubitzke
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 511617567

It is a common and rather rudimentarily understood phenomenon that a lot of polynomials with non-negative coefficients (or their symmetric decompositions), that are of importance in algebra, geometry or combinatorics, exhibit nice properties as being unimodal or real-rooted. For instance, it is known that essentially all well-studied subdivision operations on simplicial complexes, including barycentric and edgewise subdivisions give rise to real-rooted h-polynomials (under relatively weak assumptions on the original simplicial complex). In 2021, Athanasiadis could provide an explanation for this phenomenon by introducing so-called uniform triangulations, that provide a general framework for many of the well-studied subdivision operations. One of the key results is that real-rootedness of the h-polynomial of any subdivided sufficiently nice simplicial complex is implied by properties of the h-polynomials of subdivided simplices, a property he calls strongly interlacing. Together with Tzanaki he could further show that if in addition the h-vector of the original simplicial complex satisfies certain additional inequalities, then the h-polynomial of the subdivided simplicial complex has a non-negative real-rooted decomposition or is even interlacing.The main goals of this project are:Find /construct classes of uniform triangulations that are strongly interlacing. Characterize local h-vectors of uniform triangulations and their F-triangles. This should help to better understand the combinatorics of uniform triangulations.Find classes of subdivisions such that h-polynomials of sufficiently nice subdivided simplicial complexes satisfy the inequalities mentioned above. Applying these subdivisions iteratively will produce h-polynomials with a non-negative symmetric real-rooted or even interlacing decomposition.One of the classes of polynomials to be considered for all questions are s-Eulerian polynomials and s-derangement polynomials. In particular, we want to identify classes of s-Eulerian and s-derangement polynomials respectively that are h-polynomials and local h-polynomials of uniform triangulations. This is motivated by the fact that s-Eulerian and s-derangement polynomials are known to be real-rooted and the h-polynomials and local h-polynomials of the barycentric and edgewise subdivision can be realized as such. In particular, this yields classes of subdivisions that are promising candidates for being strongly interlacing.

• DFG Programme: Research Grants