Enumeration and Unimodular Equivalence of Empty Delta-Modular Simplices
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Consider a class of simplices defined by systems A x ≤ b of linear inequalities with Δ-modular matrices. A matrix is called Δ-modular, if all its rank-order sub-determinants are bounded by Δ in an absolute value. In our work we call a simplex Δ-modular, if it can be defined by a system A x ≤ b with a Δ-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 Δ is fixed, it was shown that the number of Δ-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 Δ-modular empty lattice-simplices. As the main result, assuming again that the value of the parameter Δ 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 Δ-modular, not necessarily empty, simplices.
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