2014/12/16 by José F. Fernando, Fernando, José F., Carlos Ueno +1
Mathematics · #52B10 #90C26 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities #Primary: 14P10 #Secondary: 52B55 #math.AG #msc:14P10 #msc:52B10 #msc:52B55 #msc:90C26
paper · pdf · doi:10.48550/arxiv.1412.5107
29 pages, 8 figures
openalex publication_date 2014/12/16 · arxiv created 2015/05/04 · arxiv updated 2015/05/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this work we prove constructively that the complement \mathbb Rn∖\mathcal K of an n-dimensional unbounded convex polyhedron \mathcal K⊂\mathbb Rn and the complement \mathbb Rn∖\rm Int(\mathcal K) of its interior are polynomial images of \mathbb Rn whenever \mathcal K does not disconnect \mathbb Rn. The compact case and the case of convex polyhedra of small dimension were approached by the authors in previous works. Consequently, the results of this article provide a full answer to the representation as polynomial images of Euclidean spaces of complements of convex polyhedra and its interiors. The techniques here are more sophisticated than those corresponding to the compact case and require a rational separation result for certain type of (non-compact) semialgebraic sets, that has interest by its own.