Volume polynomials and duality algebras of multi-fans

TL;DR

This paper develops a new theory connecting volume polynomials and duality algebras for multi-fans, with applications to geometry, topology, and combinatorics, and shows that certain classical theorems do not extend to multi-polytopes.

Contribution

It introduces volume polynomials and duality algebras for multi-fans, expanding the understanding of their structure and applications in various mathematical fields.

Findings

Construction of volume polynomials for multi-fans

Development of Poincare duality algebras from multi-fans

Demonstration that the $g$-theorem does not hold for multi-polytopes

Abstract

We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_\Delta$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(\Delta)$. We study the structure and properties of $V_\Delta$ and $\mathcal{A}^*(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novik--Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.