Consider a class of simplices defined by systems $A x \leq b$ of linear inequalities with $\Delta$-modular matrices. A matrix is called $\Delta$-modular, if all its rank-order sub-determinants are bounded by $\Delta$ in an absolute value. In our work we call a simplex $\Delta$-modular, if it can be defined by a system $A x \leq b$ with a $\Delta$-modular matrix $A$. And we call a simplex empty, if it contains no points with integer coordinates. In literature, a simplex is called lattice-simplex, if all its vertices have integer coordinates. And a lattice-simplex called empty, if it contains no points with integer coordinates excluding its vertices. Recently, assuming that $\Delta$ is fixed, it was shown that the number of $\Delta$-modular empty simplices modulo the unimodular equivalence relation is bounded by a polynomial on dimension. We show that the analogous fact holds for the class of $\Delta$-modular empty lattice-simplices. As the main result, assuming again that the value of the parameter $\Delta$ is fixed, we show that all unimodular equivalence classes of simplices of the both types can be enumerated by a polynomial-time algorithm. As the secondary result, we show the existence of a polynomial-time algorithm for the problem to check the unimodular equivalence relation for a given pair of $\Delta$-modular, not necessarily empty, simplices.

翻译：考虑由线性不等式系统 $A x \leq b$ 定义的一类单形，其中矩阵为 $\Delta$-模矩阵。若矩阵的所有秩阶子行列式的绝对值均以 $\Delta$ 为界，则称其为 $\Delta$-模矩阵。本文中，若单形可由带有 $\Delta$-模矩阵 $A$ 的系统 $A x \leq b$ 定义，则称该单形为 $\Delta$-模单形；若单形不含整数坐标点，则称其为空单形。文献中，若单形的所有顶点均为整数坐标点，则称其为格单形；若格单形除顶点外不含其他整数坐标点，则称其为空格单形。近期，在固定 $\Delta$ 的假设下，已有研究表明：在幺模等价关系下，$\Delta$-模空单形的数量受限于维度的多项式函数。本文证明类似结论对 $\Delta$-模空格单形类也成立。作为主要结果，在固定参数 $\Delta$ 值的假设下，我们证明两类单形的所有幺模等价类均能通过多项式时间算法枚举。作为次要结果，我们证明存在多项式时间算法，用于判断给定一对 $\Delta$-模单形（不必为空）的幺模等价关系。