Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes

Let P ₁ ,..., P _n and Q _1,..., Q _n be convex polytopes in R ⁿ such that P _i is a proper subset of Q _i . It is well-known that the mixed volume has the monotonicity property: V (P1,...,P _n ) is less than or equal to V (Q _1,..., 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 ,..., P _n whose number of isolated solutions equals the normalized volume of the convex hull of P ₁ U...U P _n . In addition, we obtain an analog of Cramer's rule for sparse polynomial systems.