Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes

Authors

Bihan Frederic, Université Savoie Mont Blanc
Ivan Soprunov, Cleveland State UniversityFollow

Document Type

Article

Publication Date

2-24-2017

Publication Title

Advances in Geometry

Abstract

Let P₁,..., P_n and Q_1,...,Q_n be convex polytopes in Rⁿ 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 P₁,..., P_n whose number of isolated solutions equals the normalized volume of the convex hull of P₁ U...U P_n . In addition, we obtain an analog of Cramer's rule for sparse polynomial systems.

Repository Citation

Frederic, Bihan and Soprunov, Ivan, "Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes" (2017). Mathematics and Statistics Faculty Publications. 279.
https://engagedscholarship.csuohio.edu/scimath_facpub/279

Version

Preprint