Algebraicity of Nash sets and of their asymmetric cobordism

Ghiloni, Riccardo; Tancredi, Alessandro

doi:10.4171/jems/672

Cited by 4 publications

(3 citation statements)

References 29 publications

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“…There is a wide literature devoted to improvements and extensions of Nash-Tognoli theorem. For this topic, we refer the reader to the books [2, Chapter II], [8,Chapter 14], [22,Chapter 6], the survey [20, Section 2], the recent papers [6,13,21] and the numerous references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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The topology of real algebraic sets with isolated singularities is determined by the field of rational numbers

Ghiloni¹,

Savi²

2023

Preprint

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We prove that every real algebraic set V ⊂ R n with isolated singularities is homeomorphic to a set V ′ ⊂ R m that is Q-algebraic in the sense that V ′ is defined in R m by polynomial equations with rational coefficients. The homeomorphism φ : V → V ′ we construct is semialgebraic, preserves nonsingular points and restricts to a Nash diffeomorphism between the nonsingular loci. In addition, we can assume that V ′ has a codimension one subset of rational points. If m is sufficiently large, we can also assume that V ′ ⊂ R m is arbitrarily close to V ⊂ R n ⊂ R m , and φ extends to a semialgebraic homeomorphism from R m to R m . A first consequence of this result is a Q-version of the Nash-Tognoli theorem: Every compact smooth manifold admits a Q-algebraic model. Another consequence concerns the open problem of making Nash germs Q-algebraic: Every Nash set germ with an isolated singularity is semialgebraically equivalent to a Q-algebraic set germ.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The reader observes that condition (13) remains valid if we replace U a with a smaller open neighborhood of a in R n . In addition, it is evident that the disjoint union of finitely many Q-nice real algebraic subsets of R n is again a Q-nice real algebraic subset of R n .…”

Section: Q-algebraic Approximations à La Akbulut-kingmentioning

confidence: 99%

The topology of real algebraic sets with isolated singularities is determined by the field of rational numbers

Ghiloni¹,

Savi²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…There is a wide literature devoted to improvements and extensions of Nash-Tognoli theorem. We refer the interested reader to the books [AK92, Chapter II], [BCR98, Chapter 14], [Man20, Chapter 6], the survey [Kol17, Section 2], the papers [CS92,Kuc11], the more recent ones [Ben,GT17] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Embedded $\mathbb{Q}$-desingularization of real Schubert varieties and application to the relative $\mathbb{Q}$-algebrization of nonsingular algebraic sets

Savi¹

2023

Preprint

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A real algebraic set V ⊂ R n is said to be a Q-nonsingular Q-algebraic set if it is described, both globally and locally, by polynomials with rational coefficients. We prove that every nonsingular real algebraic set V ⊂ R n with nonsingular algebraic subsetsA key result in the proof is the description of Z/2Z-homological cycles of real Grassmannian manifolds by Q-nonsingular Q-algebraic representatives via an explicit desingularization of real embedded Schubert varieties.

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Differentiable approximation of continuous semialgebraic maps

Fernando

Ghiloni

2019

Sel. Math. New Ser.

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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.

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Algebraicity of Nash sets and of their asymmetric cobordism

Cited by 4 publications

References 29 publications

The topology of real algebraic sets with isolated singularities is determined by the field of rational numbers

The topology of real algebraic sets with isolated singularities is determined by the field of rational numbers

Embedded $\mathbb{Q}$-desingularization of real Schubert varieties and application to the relative $\mathbb{Q}$-algebrization of nonsingular algebraic sets

Differentiable approximation of continuous semialgebraic maps

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