Classification of triples of lattice polytopes with a given mixed volume

Abstract

We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume $m$ in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed $m$. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals.

26 pages, additional appendix of 34 pages

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• Canonical toric Fano threefolds
• Galois theory for general systems of polynomial equations
• Lattice 3-polytopes with six lattice points
• Lattice 3-polytopes with few lattice points
• Enumeration of lattice 3-polytopes by their number of lattice points
• The complete classification of empty lattice $4$-simplices
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• Clean Lattice Tetrahedra
• Classification of triples of lattice polytopes with a given mixed volume
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• Normal forms of convex lattice polytopes
• Multivariate Abel-Ruffini
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Cited by in corpus (2)

• Classification of triples of lattice polytopes with a given mixed volume
• Polyhedra of small relative mixed volume