Title: Criteria for strict monotonicity of the mixed volume of convex polytopes,
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.