Enumeration and Unimodular Equivalence of Empty Delta-Modular Simplices

Abstract: 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.