Classification of Triples of Lattice Polytopes with a Given Mixed Volume
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Classification of Triples of Lattice Polytopes with a Given Mixed Volume Gennadiy Averkov1 · Christopher Borger2
· Ivan Soprunov3
Received: 31 January 2019 / Revised: 21 February 2020 / Accepted: 23 August 2020 © The Author(s) 2020
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. Keywords Bernstein–Khovanskii–Kouchnirenko theorem · Classification · Lattice polytope · Mixed volume · Newton polytope · Sparse polynomial systems Mathematics Subject Classification 14M25 · 52B20 · 52A39 · 52A38 · 52B10
Editor in Charge: Kenneth Clarkson Gennadiy Averkov [email protected] Christopher Borger [email protected] Ivan Soprunov [email protected] 1
Fakultät 1, BTU Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany
2
Faculty of Mathematics, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
3
Department of Mathematics and Statistics, Cleveland State University, 2121 Euclid Ave, Cleveland, OH 44115, USA
123
Discrete & Computational Geometry
1 Introduction 1.1 Formulation of the Problem and Previous Results In the last decade, there has been an increased interest in the algorithmic theory of lattice polytopes, which is motivated by applications in algebra, algebraic geometry, combinatorics, and optimization (see, for example, [3,4,6,8–12,19–21,23]). So far, a special emphasis has been put on computer-assisted enumeration results, which are important from different perspectives. On the one hand, one can carry out enumeration to gather concrete data and then make and verify hypotheses based on these data. On the other hand, proving results on lattice polytopes frequently requires to handle special cases, which are only finitely many but are hard to determine without algorithmic assistance. Most notably, some structural classification results for lattice polytopes hold up to finitely many exceptional situations and, in this case, one is interested in describing the exceptional situations by means of enumeration in order to accomplish a classification. Thus, the structural and algorithmic theory are intertwined and constantly influence each other. Evaluation of new data established using a computer-assisted search leads to new theoretical questions, while new theoretical results (in particular, finiteness results) suggest new enumeration tasks. Our point of departure is the classical Bernstein–Khovanskii–K
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