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.
Highlights
1.1 Formulation of the Problem and Previous ResultsIn 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
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
In this paper we present an algorithmic approach to Classification Problem 1.3 when m and d are given and d ≤ 3
Summary
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, [3,4,6,8,9,10,11,12,19,20,21,23]). By Theorem 1.1, if the vector (c1,(0,0), c1,(1,0), c1,(2,0), c1,(0,1), c2,(0,0), c2,(1,0), c2,(0,1), c2,(0,2)) ∈ C8 of all coefficients of the polynomials f1 and f2 is generic, the system f1 = f2 = 0 has exactly four solutions in (C \ {0}), because the normalized mixed volume V (P1, P2) of P1 and P2 equals 4. Pd ) of lattice polytopes whose normalized mixed volume equals m The solution of this problem has been known only in the following cases:. In [14] Esterov showed that the number of irreducible d-tuples of lattice polytopes is finite up to equivalence for a fixed mixed volume and dimension (see Theorem 2.4 below). There is a natural partial ordering on the set of equivalence classes of d-tuples of lattice polytopes with a given mixed volume m, induced by inclusion.
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