# Criteria for strict monotonicity of the mixed volume of convex polytopes

## TL;DR

This paper establishes criteria for strict monotonicity of mixed volume of convex polytopes, linking geometric face structures to solution counts of sparse polynomial systems, and introduces an analog of Cramer's rule.

## Contribution

It provides new geometric criteria for strict mixed volume inequalities and characterizes when sparse polynomial systems attain maximal solution counts.

## Key findings

- Criteria for strict inequality based on faces and subdivisions.
- Characterization of polynomial systems with maximal solutions.
- An analog of Cramer's rule for sparse systems.

## Abstract

Let $P_1,\dots, P_n$ and $Q_1,\dots, Q_n$ be convex polytopes in $\mathbb{R}^n$ such that $P_i\subset Q_i$. It is well-known that the mixed volume has the monotonicity property: $V(P_1,\dots,P_n)\leq V(Q_1,\dots,Q_n)$. We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes $P_1,\dots,P_n$ whose number of isolated solutions equals the normalized volume of the convex hull of $P_1\cup\dots\cup P_n$. In addition, we obtain an analog of Cramer's rule for sparse polynomial systems.