Criteria for strict monotonicity of the mixed volume of convex polytopes

Abstract

Abstract Let P 1, …, Pn and Q 1, …, Qn be convex polytopes in ℝ n with Pi ⊆ Qi . It is well-known that the mixed volume is monotone: V(P 1, …, Pn ) ≤ V(Q 1, …, 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 P 1, …, Pn whose number of isolated solutions equals the normalized volume of the convex hull of P 1 ∪ … ∪ Pn . In addition, we obtain an analog of Cramer’s rule for sparse polynomial systems.