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Abstract:
The elementary closure $P'$ of a polyhedron $P$ is the intersection
of $P$ with all its Gomory-Chvátal cutting planes.
$P'$ is a rational polyhedron provided that $P$ is rational. The
known bounds for the number of inequalities defining $P'$ are
exponential, even in fixed dimension.
We show that the number of inequalities needed to describe the
elementary closure of a rational polyhedron is polynomially bounded
in fixed dimension.
If $P$ is a simplicial cone, we construct
a polytope $Q$, whose integral elements correspond to cutting planes
of $P$. The vertices of
the integer hull $Q_I$ include the facets of $P'$.
A polynomial upper bound on their number can be obtained by
applying a result of Cook et al.
Finally, we present a polynomial algorithm in varying dimension,
which computes cutting planes for a simplicial cone that
correspond to vertices of $Q_I$.
