Veronese sections and interlacing matrices of polynomials and formal power series

TL;DR

This paper introduces the concept of fully interlacing matrices of formal power series, extending classical polynomial interlacing, and demonstrates their properties and applications in triangulations of simplicial complexes.

Contribution

It defines fully interlacing matrices of power series, proves their stability under various operations, and connects these results to geometric applications in simplicial complex triangulations.

Findings

Fully interlacing matrices preserve interlacing under matrix operations.

The concept generalizes and unifies previous interlacing results.

Applications to uniform triangulations of simplicial complexes are provided.

Abstract

The concept of a fully interlacing matrix of formal power series with real coefficients is introduced. This concept extends and strengthens that of an interlacing sequence of real-rooted polynomials with nonnegative coefficients, in the special case of row and column matrices. The fully interlacing property is shown to be preserved under matrix products, flips across the reverse diagonal and Veronese sections of the power series involved. These results and their corollaries generalize, unify and simplify several results which have previously appeared in the literature. An application to the theory of uniform triangulations of simplicial complexes is included.