On Nash images of Euclidean spaces

Ueno²

2023

J. Math. Soc. Japan

In this work we approach the problem of determining which (compact) semialgebraic subsets of R n are images under polynomial maps f : R m → R n of the closed unit ball Bm centered at the origin of some Euclidean space R m and that of estimating (when possible) which is the smallest m with this property. Contrary to what happens with the images of R m under polynomial maps, it is quite straightforward to provide basic examples of semialgebraic sets that are polynomial images of the closed unit ball. For instance, simplices, cylinders, hypercubes, elliptic, parabolic or hyperbolic segments (of dimension n) are polynomial images of the closed unit ball in R n .The previous examples (and other basic ones proposed in the article) provide a large family of 'n-bricks' and we find necessary and sufficient conditions to guarantee that a finite union of 'n-bricks' is again a polynomial image of the closed unit ball either of dimension n or n + 1. In this direction, we prove: A finite union S of n-dimensional convex polyhedra is the image of the n-dimensional closed unit ball Bn if and only if S is connected by analytic paths.The previous result can be generalized using the 'n-bricks' mentioned before and we show: If S1, . . . , S ⊂ R n are 'n-bricks', the union S := i=1 Si is the image of the closed unit ball Bn+1 of R n+1 under a polynomial map f : R n+1 → R n if and only if S is connected by analytic paths.

Section: Finite Unions Of Convex Polyhedramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On polynomial images of a closed ball

Ueno²

2023

J. Math. Soc. Japan

“…A Nash function on an open semialgebraic set U ⊂ M is a semialgebraic smooth function on U . In [Fe2] the first author solves Problem 1.2 and provides a full characterization of the semialgebraic subsets of R m that are images under a Nash map on some Euclidean space. A natural alternative approach is to consider regulous maps, which are closer than Nash maps to regular maps but seem to be less rigid than the latter [K1, K2].…”

Section: Introductionmentioning

confidence: 99%

On regulous and regular images of Euclidean spaces

The Quarterly Journal of Mathematics

Fichou

Quarez

et al. 2018

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.

Differentiable approximation of continuous semialgebraic maps

Ghiloni

2019

Sel. Math. New Ser.

Self Cite

In this work we approach the problem of approximating uniformly continuous semialgebraic maps f : S Ñ T from a compact semialgebraic set S to an arbitrary semialgebraic set T by semialgebraic maps g : S Ñ T that are differentiable of class C ν for a fixed integer ν ě 1. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For ν " 1 we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For ν ě 2 we obtain density results in the two following relevant situations: either T is compact and locally C ν semialgebraically equivalent to a polyhedron, for instance when T is a compact polyhedron; or T is an open semialgebraic subset of a Nash set, for instance when T is a Nash set. Our density results are based on a recent C 1 -triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples.