Induced Lorentzian and volume polynomials

TL;DR

This paper demonstrates that the counts of specific panels covering topics form a Lorentzian polynomial, using an inducing operator that preserves Lorentzian properties and relates to induced (poly)matroids.

Contribution

It introduces an inducing operator that preserves Lorentzian polynomials and applies it to volume polynomials related to topic coverage panels.

Findings

Counts of topic-covering panels form Lorentzian polynomials.

The inducing operator preserves Lorentzian and volume polynomials.

Provides a new connection between polynomials, matroids, and combinatorial coverage problems.

Abstract

Suppose one has a party of $m$ people, whose expertise collectively covers $n$ topics. Given a subset $T$ of the topics, one wishes to form a panel of $|T|$ people from the party such that $T$ can be covered by assigning a distinct topic to each panel member with the expertise. We show that the numbers of such panels, as $T$ varies, form a Lorentzian polynomial. We achieve this by showing that a certain linear operator on polynomials, which we call the ``inducing operator'' for its connection to induced (poly)matroids, preserves Lorentzian polynomials and realizable volume polynomials.