On the complements of 3-dimensional convex polyhedra as polynomial images of ℝ<sup>3</sup>

Fernando, José F.; Ueno, Carlos

doi:10.1142/s0129167x14500712

Cited by 9 publications

(10 citation statements)

References 13 publications

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“…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%

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%

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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Section: Introductionmentioning

confidence: 99%

“…Even though, we have approached the representation as polynomial or regular images of ample families of n-dimensional semialgebraic sets whose boundaries are piecewise linear. We have focused on: convex polyhedra, their interiors, their exteriors and the closure of their exteriors [FGU1,FU1,FU2,U2]. The proofs are constructive but the arguments are developed ad hoc.…”

Section: Introductionmentioning

confidence: 99%

“…The open ones deserve a special attention in connection with the real Jacobian Conjecture [J1, J2, P]. The interest of polynomial (and also regular) images arises because there exist certain problems in Real Algebraic Geometry that can be reduced for such sets to the case S " R n (see [FU1,FU2]). Examples of such problems are: ‚ Optimization of polynomial (and/or regular) functions on S, ‚ Characterization of the polynomial (or regular functions) that are positive semidefinite on S (Hilbert's 17th problem and Positivstellensatz), ‚ Computation of trajectories inside S parametrizable by polynomial (or regular) maps.…”

Section: Introductionmentioning

confidence: 99%

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