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.

中文翻译：

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具有给定混合体积的格多面体三元组的分类

我们提出了一种对具有给定混合体积 m 的晶格多胞体三元组在维度 3 中进行分类的算法。众所周知，分类可以简化为所谓的不可约三元组的枚举，对于固定的 m，三元组的数量是有限的. 按照这个算法，我们枚举了归一化混合体积的所有不可约三元组，最多为 4，它们是最大包含。这产生了通用三变量稀疏多项式系统的分类，在复杂环面上最多有 4 个解，最多可达变量的单项式变化。根据 Esterov 的最新结果，这导致了所有可通过根求解的通用三变量稀疏多项式系统的描述。