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07 February 2025
 
  » arxiv » 1508.0195

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Dimension groups and linear Diophantine inequalities
David Handelman ; Damien Roy ;
Date 2 Aug 2015
AbstractSolvability of systems of linear diophantine inequalities is related to dimension group properties, yielding necessary and sufficient conditions. Specifically, if $H$ is a subgroup of Euclidean space, for every $h$ in $H$ and positive integer $m$, for every $epsilon > 0$, we can solve for $x$ in $H$ the inequalities $h-mx > -epsilon$ coordinatewise, iff for $F$ defined as the smallest face of the standard simplex (in the dual of the Euclidean space) that contains the positive linear functionals killing $H$, for all $f$ in $F$, the subgroup of the reals, $f(H)$, is either zero or dense.
Source arXiv, 1508.0195
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