On Complements of Convex Polyhedra as Polynomial and Regular Images of ℝn

Fernando, José F.; Ueno, Carlos

doi:10.1093/imrn/rnt112

Cited by 16 publications

(23 citation statements)

References 14 publications

(27 reference statements)

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“…Thus, any improvement in the complexity of the involved polynomial maps would lead to a better output, and this also applies to the first step of the process, directly connected to the open quadrant problem. A similar argument applies to prove that the complement in R n of an n-dimensional convex polyhedron is a polynomial image of R n (Fernando and Ueno, 2014a), which uses an inductive process starting with the complement of the open orthant. In order to represent this complement of the orthant as a polynomial image we need first to obtain the complement of the (closed) quadrant in R 2 as a polynomial image, and this is achieved by applying to the open quadrant (considered in C) the polynomial map z → z 3 .…”

Section: Characterize Geometrically the Images Of Polynomial Maps Betmentioning

confidence: 99%

“…Examples of these problems arise in Optimization or in the search for Positivstellensätze certificates (Fernando and Gamboa, 2006;Fernando and Ueno, 2014a).…”

Section: Characterize Geometrically the Images Of Polynomial Maps Betmentioning

confidence: 99%

“…On the other hand, we have described how to obtain constructively notable families of semialgebraic sets as images of polynomial or regular maps. In particular, we have focused our attention in convex polyhedra, their interiors and their complementaries (Fernando et al, 2011;Fernando and Ueno, 2014a;Fernando and Ueno, 2014b;Ueno, 2012).…”

Section: Characterize Geometrically the Images Of Polynomial Maps Betmentioning

confidence: 99%

See 2 more Smart Citations

A short proof for the open quadrant problem

Fernando

Ueno

2017

Journal of Symbolic Computation

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In 2003 it was proved that the open quadrant Q := {x > 0, y > 0} of R 2 is a polynomial image of R 2 . This result was the origin of an ulterior more systematic study of polynomial images of Euclidean spaces. In this article we provide a short proof of the previous fact that does not involve computer calculations, in contrast with the original one. The strategy here is to represent the open quadrant as the image of a polynomial map that can be expressed as the composition of three simple polynomial maps whose images can be easily understood.

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Section: Characterize Geometrically the Images Of Polynomial Maps Betmentioning

confidence: 99%

“…Examples of these problems arise in Optimization or in the search for Positivstellensätze certificates (Fernando and Gamboa, 2006;Fernando and Ueno, 2014a).…”

Section: Characterize Geometrically the Images Of Polynomial Maps Betmentioning

confidence: 99%

Section: Characterize Geometrically the Images Of Polynomial Maps Betmentioning

confidence: 99%

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A short proof for the open quadrant problem

Fernando

Ueno

2017

Journal of Symbolic Computation

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“…• To prove (constructively) that large families of semialgebraic sets with piecewise linear boundary (convex polyhedra, their interiors, their complements and the interiors of their complements) are either polynomial or regular images of Euclidean spaces [FGU1,FGU4,FU1,FU2,FU5,U1,U2].…”

Section: Introductionmentioning

confidence: 99%

On regulous and regular images of Euclidean spaces

Fernando

Fichou

Quarez

et al. 2018

The Quarterly Journal of Mathematics

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In this work we compare the semialgebraic subsets that are images of regulous maps with those that are images of regular maps. Recall that a map f : R n → R m is regulous if it is a rational map that admits a continuous extension to R n . In case the set of (real) poles of f is empty we say that it is regular map. We prove that if S ⊂ R m is the image of a regulous map f : R n → R m , there exists a dense semialgebraic subset T ⊂ S and a regular map g : R n → R m such that g(R n ) = T. In case dim(S) = n, we may assume that the difference S \ T has codimension ≥ 2 in S. If we restrict our scope to regulous maps from the plane the result is neat: if f : R 2 → R m is a regulous map, there exists a regular map g : R 2 → R m such that Im(f ) = Im(g). In addition, we provide in the Appendix a regulous and a regular map f, g : R 2 → R 2 whose common image is the open quadrant Q := {x > 0, y > 0}. These maps are much simpler than the best known polynomial maps R 2 → R 2 that have the open quadrant as their image.

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“…Concerning the representation of unbounded polygons as polynomial images see [U3]. In [FU2] we prove that the complement R n \ K of a convex polyhedron K ⊂ R n that does not disconnect R n and the complement R n \ Int K of its interior are regular images of R n . If K is moreover bounded or has dimension d < n, then R n \ K and R n \ Int K are polynomial images of R n .…”

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confidence: 99%