On complements of convex polyhedra as polynomial and regular images of $\R^n$

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Fernando Galván, José Francisco and Ueno, Carlos (2014) On complements of convex polyhedra as polynomial and regular images of $\R^n$. International Mathematics Research Notices, 25 (7). pp. 5084-5123. ISSN 1073-7928

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Official URL: http://imrn.oxfordjournals.org/content/2014/18/5084

URLURL Type
http://arxiv.org/abs/1212.1813UNSPECIFIED

Abstract

In this work we prove constructively that the complement $\R^n\setminus\pol$ of a convex polyhedron $\pol\subset\R^n$ and the complement $\R^n\setminus\Int(\pol)$ of its interior are regular images of $\R^n$. If $\pol$ is moreover bounded, we can assure that $\R^n\setminus\pol$ and $\R^n\setminus\Int(\pol)$ are also polynomial images of $\R^n$. The construction of such regular and polynomial maps is done by double induction on the number of \em facets \em (faces of maximal dimension) and the dimension of $\pol$; the careful placing (\em first \em and \em second trimming positions\em) of the involved convex polyhedra which appear in each inductive step has interest by its own and it is the crucial part of our technique.

Item Type: Article Polynomial and regular maps and images, convex polyhedra, first and second trimming positions, trimming maps, optimization, Positivstellens¨atze Sciences > Mathematics > Algebraic geometry 29172 11 Mar 2015 09:54 11 Mar 2015 09:54