On regulous and regular images of Euclidean spaces

Fernando, José F.; Fichou, Goulwen; Quarez, Ronan; Ueno, Carlos

doi:10.1093/qmath/hay027

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…Other types of maps (like Nash, continuous rational, etc.) have been already considered to represent semialgebraic sets as images of affine spaces [Fe2,FFQU]. A complete solution to Problem 1.8 seems far, but we have developed significant progresses:…”

Section: Introductionmentioning

confidence: 99%

On polynomial images of a closed ball

Fernando

Ueno²

2023

J. Math. Soc. Japan

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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.

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Section: Introductionmentioning

confidence: 99%

On polynomial images of a closed ball

Fernando

Ueno²

2023

J. Math. Soc. Japan

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“…The concise name ‘regulous’ was coined by Fichou, Huisman, Mangolte and Monnier [FHMM16]. Since the publication of [Kuc09] in 2009 several mathematicians have devoted their attention to regulous maps (see [BKVV13, Cza19, FFQU18, FHMM16, FMQ17, FMQ20, FMQ21b, FMQ21a, KKK18, KN15, Kuc09, Kuc13, Kuc14a, Kuc14b, Kuc15, Kuc16a, Kuc16b, Kuc20, KK16, KK17, KZ18, KK18a, KK18b, Mon18, Zie16, Zie18] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Approximation and homotopy in regulous geometry

Kucharz

2023

Compositio Math.

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Let $X$, $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$-regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$, we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$-regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$-regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$-dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$-regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$.

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On regulous and regular images of Euclidean spaces

Cited by 2 publications

References 17 publications

On polynomial images of a closed ball

On polynomial images of a closed ball

Approximation and homotopy in regulous geometry

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