On the Substitution Theorem for Rings of Semialgebraic Functions

Fernando, José F.

doi:10.1017/s1474748014000206

Cited by 11 publications

(9 citation statements)

References 26 publications

(54 reference statements)

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“…The existence of a (continuous) semialgebraic extension is never an obstacle: any locally compact semialgebraic set M ⊂ R m is the semialgebraic retract of a small open semialgebraic neighborhood V ⊂ R m of M , see [DK1,Thm.1]. For non-locally compact semialgebraic sets even the existence of a semialgebraic extension map is a delicate point [Fe2]. As we see below, if f : M → R is a semialgebraic function, there exists a semialgebraic embedding j : M ֒→ R p and a semialgebraic function…”

Section: Rings Of S R -Functionsmentioning

confidence: 99%

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Rings of differentiable semialgebraic functions

Baro,

Fernando,

Gamboa

2019

Preprint

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In this work we analyze the main properties of the Zariski and maximal spectra of the ring S r (M ) of differentiable semialgebraic functions of class C r on a semialgebraic set M ⊂ R m . Denote S 0 (M ) the ring of semialgebraic functions on M that admit a continuous extension to an open semialgebraic neighborhood of M in Cl(M ). This ring is the real closure of S r (M ). If M is locally compact, the ring S r (M ) enjoys a Lojasiewicz's Nullstellensatz, which becomes a crucial tool. Despite S r (M ) is not real closed for r ≥ 1, the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring S 0 (M ). In addition, the quotients of S r (M ) by its prime ideals have real closed fields of fractions, so the ring S r (M ) is close to be real closed. The missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of S r (M ) and S 0 (M ) guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring S r (M ) is a Gelfand ring and its Krull dimension is equal to dim(M ). We also show similar properties for the ring S r * (M ) of differentiable bounded semialgebraic functions. In addition, we confront the ring S ∞ (M ) of differentiable semialgebraic functions of class C ∞ with the ring N (M ) of Nash functions on M .

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Section: Rings Of S R -Functionsmentioning

confidence: 99%

“…Comparison between S 0 (M ) and S(M ). In [Fe2,Cor.6] it is proved that if M is a 2-dimensional semialgebraic set such that the germ M x is connected for each x ∈ Cl(M ), then S(M ) = S 0 (M ). The following result is the 2-dimensional version of Theorem 1.1 and generalizes [Fe2,Cor.6…”

Section: Dmentioning

confidence: 99%

Rings of differentiable semialgebraic functions

Baro,

Fernando,

Gamboa

2019

Preprint

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“…Thus, m α := p α is the maximal ideal of S(M ) satisfying m α ∩ S * (M ) ⊂ m * α . The last part of the statement follows from [Fe2,(1.B.2)].…”

Section: Cmentioning

confidence: 99%

On nuclearity of embeddings between Besov spaces

Cobos

Domnguez

Khn

2018

Journal of Approximation Theory

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In this work we analyze some topological properties of the remainder ∂M := β * s M \ M of the semialgebraic Stone-Cěch compactification β * s M of a semialgebraic set M ⊂ R m in order to 'distinguish' its points from those of M. To that end we prove that the set of points of β * s M that admit a metrizable neighborhood in β * s M equals M lc ∪ (Cl β * s M (M ≤1) \ M ≤1) where M lc is the largest locally compact dense subset of M and M ≤1 is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets ∂M and ∂M of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ∂M and that the differences ∂M \ ∂M and ∂M \ ∂M are also dense subsets of ∂M. It holds moreover that all the points of ∂M have countable systems of neighborhoods in β * s M .

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“…The authors consider there the case of the oriented blow-up of a real analytic manifold M with center a closed real analytic submanifold N whose vanishing ideal inside M is finitely generated (this happens for instance if N is compact). In [Fe2,§3] we present a similar construction in the semialgebraic setting, which is used to 'appropriately embed' semialgebraic sets in Euclidean spaces. The following result, whose proof makes use of the drilling blow-up and which will be a key tool for our purposes, will allow to erase a closed Nash submanifold from a Nash manifold (see Figure 5).…”

Section: A Main Ingredient: the Drilling Blow-upmentioning

confidence: 99%

On Nash images of Euclidean spaces

Fernando

2018

Advances in Mathematics

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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.

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On the Substitution Theorem for Rings of Semialgebraic Functions

Cited by 11 publications

References 26 publications

Rings of differentiable semialgebraic functions

Rings of differentiable semialgebraic functions

On nuclearity of embeddings between Besov spaces

On Nash images of Euclidean spaces

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