Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes

Let P1,..., Pn and Q1,...,Qn be convex polytopes in Rn such that Pi is a proper subset of Qi . It is well-known that the mixed volume has the monotonicity property: V (P1,...,Pn) is less than or equal to V (Q1,...,Qn) . 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 P1,..., Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 U...U Pn . In addition, we obtain an analog of Cramer\u27s rule for sparse polynomial systems