On the Set of Points at Infinity of a Polynomial Image of $${\mathbb R}^n$$ R n

Journal of Symbolic Computation

Ueno

2017

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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.

Section: Characterize Geometrically the Images Of Polynomial Maps Betmentioning

confidence: 99%

A short proof for the open quadrant problem

Journal of Symbolic Computation

Ueno

2017

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“…A survey concerning this topic, which provides the reader a global idea of the state of the art, can be found in [FGU4]. Articles [FGU2,FU3] are of different nature. In them we find new obstructions for a semialgebraic subset of R n to be either a polynomial or a regular image of R m .…”

Section: Introductionmentioning

confidence: 99%

On Nash images of Euclidean spaces

Advances in Mathematics

2018

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In this work we characterize the subsets of R n that are images of Nash maps f : R m → R n . We prove Shiota's conjecture and show that a subset S ⊂ R n is the image of a Nash map f : R m → R n if and only if S is semialgebraic, pure dimensional of dimension d ≤ m and there exists an analytic path α : [0, 1] → S whose image meets all the connected components of the set of regular points of S. Two remarkable consequences are the following:(1) pure dimensional irreducible semialgebraic sets of dimension d with arc-symmetric closure are Nash images of R d ; and (2) semialgebraic sets are projections of irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces.

“…By obtaining general conditions that must satisfy a semialgebraic subset S ⊂ R m which is either a polynomial or a regular image of R n (see [FG2,FU1,U1] for further details). The most remarkable one states that the set of infinite points of a polynomial image of R n is connected.…”

Section: Introductionmentioning

confidence: 99%

“…We feel very far from solving the problem stated above in its full generality, but we have developed significant progresses in two ways: General conditions. By obtaining general conditions that must satisfy a semialgebraic subset S ⊂ R m which is either a polynomial or a regular image of R n (see [FG2,FU1,U1] for further details). The most remarkable one states that the set of infinite points of a polynomial image of R n is connected.…”

Section: Introductionmentioning

confidence: 99%

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

International Mathematics Research Notices

Ueno

2013

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Abstract. In this work we prove constructively that the complement R n \K of a convex polyhedron K ⊂ R n and the complement R n \ Int(K) of its interior are regular images of R n . If K is moreover bounded, we can assure that R n \ K and R n \ Int(K) are also polynomial images of R n . The construction of such regular and polynomial maps is done by double induction on the number of facets (faces of maximal dimension) and the dimension of K; the careful placing (first and second trimming positions) 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.