Bounded super real closed rings

Tressl, Marcus

doi:10.1017/cbo9780511778421.011

Cited by 6 publications

(1 citation statement)

References 2 publications

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“…The rings

S^{0true ⋄} (M)

are particular cases of the so‐called real closed rings introduced by Schwartz in the '80s of the last century . The theory of real closed rings has been deeply developed until now in a fruitful attempt to establish new foundations for semi‐algebraic geometry with relevant interconnections to model theory, see the results of Cherlin–Dickmann , , Schwartz , , , , Schwartz with Prestel, Madden and Tressl , , and Tressl –. We refer the reader to for a ring theoretic analysis of the concept of real closed ring.…”

Section: Introductionmentioning

confidence: 99%

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Fernando

2016

Mathematische Nachrichten

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Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)1:Spec(scriptS(N))→Spec(scriptS(M)) and Spec(j)2:Spec(scriptS*(N))→Spec(scriptS*(M)) induced by the inclusion j:N↪M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset WM of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion frakturiM:Spec(scriptS(M))↪Spec(scriptS*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)2:Spec(scriptS*(N))→Spec(scriptS*(M)). If we denote Z:=prefixClSpec(S*(M))(M∖N), it holds that the restriction map Spec(j)2|:prefixSpec(scriptS*(N))∖Spec(j)2−1(Z)→Spec(scriptS*(M))∖Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)2 at the points of Z. The size of the fibers of prime ideals “close” to the complement Y:=M∖N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)2 is surjective and the generic fiber of a prime ideal p∈Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)2−1(frakturp) is a finite set for p∈Z. If such is the case, our procedure allows us to compute the size s of Spec(j)2−1(frakturp). If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in frakturp.

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“…The rings

S^{0true ⋄} (M)

Section: Introductionmentioning

confidence: 99%

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Fernando

2016

Mathematische Nachrichten

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Rings of differentiable semialgebraic functions

Baro,

Fernando,

Gamboa

2024

Sel. Math. New Ser.

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On the Substitution Theorem for Rings of Semialgebraic Functions

Fernando

2014

J. Inst. Math. Jussieu

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Abstract. Let R ⊂ F be an extension of real closed fields and S(M, R) the ring of (continuous) semialgebraic functions on a semialgebraic set M ⊂ R n . We prove that every R-homomorphism ϕ : S(M, R) → F is essentially the evaluation homomorphism at a certain point p ∈ F n adjacent to the extended semialgebraic set MF . This type of result is commonly known in Real Algebra as Substitution Theorem. In case M is locally closed, the results are neat while the non locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest on their own. We afford the same problem for the ring of bounded (continuous) semialgebraic functions getting results of a different nature.

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Bounded super real closed rings

Cited by 6 publications

References 2 publications

On the size of the fibers of spectral maps induced by semialgebraic embeddings

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Rings of differentiable semialgebraic functions

On the Substitution Theorem for Rings of Semialgebraic Functions

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