Criteria for strict monotonicity of the mixed volume of convex polytopes
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Frédéric Bihan
Abstract
Let P1, …, Pn and Q1, …, Qn be convex polytopes in ℝn with Pi ⊆ Qi. It is well-known that the mixed volume is monotone: V(P1, …, Pn) ≤ 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 ∪ … ∪ Pn. In addition, we obtain an analog of Cramer’s rule for sparse polynomial systems.
Communicated by: M. Joswig
Acknowledgements
This project began at the Einstein Workshop on Lattice Polytopes at Freie Universität Berlin in December 2016. We are grateful to Mónica Blanco, Christian Haase, Benjamin Nill, and Francisco Santos for organizing this wonderful event and to the Harnack Haus for their hospitality. We thank Gennadiy Averkov and Christian Haase for fruitful discussions. Finally, we are thankful to the anonymous referee for their comments and suggestions that led to a substantial improvement of the exposition.
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Articles in the same Issue
- Frontmatter
- A Batyrev type classification of ℚ-factorial projective toric varieties
- Blocking sets of certain line sets related to a hyperbolic quadric in PG(3, q)
- Affine-compact functors
- Limit points of the branch locus of 𝓜g
- Criteria for strict monotonicity of the mixed volume of convex polytopes
- Principal curvatures and parallel surfaces of wave fronts
- Nodal curves with a contact-conic and Zariski pairs
- Corrigendum to the paper “On surfaces with two apparent double points”
Articles in the same Issue
- Frontmatter
- A Batyrev type classification of ℚ-factorial projective toric varieties
- Blocking sets of certain line sets related to a hyperbolic quadric in PG(3, q)
- Affine-compact functors
- Limit points of the branch locus of 𝓜g
- Criteria for strict monotonicity of the mixed volume of convex polytopes
- Principal curvatures and parallel surfaces of wave fronts
- Nodal curves with a contact-conic and Zariski pairs
- Corrigendum to the paper “On surfaces with two apparent double points”
