Geometry of Subanalytic and Semialgebraic Sets

Shiota, Masahiro

doi:10.1007/978-1-4612-2008-4

Cited by 140 publications

(194 citation statements)

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“…A function f : R n → R is semi-algebraic if epi f ⊂ R n+1 is a semi-algebraic set. For an extensive discussion on semi-algebraic geometry, see the monographs of Basu-Pollack-Roy [3], Lou van den Dries [26], and Shiota [25]. For a quick survey, see the article of van den Dries-Miller [27] and the surveys of Coste [9,10].…”

Section: Semi-algebraic Geometrymentioning

confidence: 99%

Quadratic growth and critical point stability of semi-algebraic functions

Ioffe

2014

Math. Program.

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Section: Semi-algebraic Geometrymentioning

confidence: 99%

Quadratic growth and critical point stability of semi-algebraic functions

Ioffe

2014

Math. Program.

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“…(In the case of tame functions, "finiteness" of critical points should be replaced by "local isolation" in Proposition 4.3 and Corollaries 4.4, 5.8, 5.9.) In particular, our results hold for globally subanalytic functions, discussed in [36]. For a quick introduction to these concepts in an optimization context, see [22].…”

Section: Introductionmentioning

confidence: 63%

“…See the monographs of Basu-Pollack-Roy [1], Lou van den Dries [37], and Shiota [36]. For a quick survey, see the article of van den Dries-Miller [38] and the surveys of Coste [17,16].…”

Section: Semi-algebraic Geometrymentioning

confidence: 99%

Semi-algebraic functions have small subdifferentials

Lewis

2012

Math. Program.

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“…In particular, we see why there are not non-constant periodic functions. This finiteness quality is an old pet of Real Geometry, and is perfectly expressed by the modern notion of o-minimal structure (see [vdD]); the book [Sh3] goes along the same ideas.…”

Section: The Notion Of Nash Functionmentioning

confidence: 98%

Global Problems on Nash Functions.

Shiota

Coste²,

Sancho

2004

Rev. Mat. Complut.

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This is a survey on the history of and the solutions to the basic global problems on Nash functions, which have been only recently solved, namely: separation, extension, global equations, Artin-Mazur description and idempotency, also noetherianness. We discuss all of them in the various possible contexts, from manifolds over the reals to real spectra of arbitrary commutative rings.Nash functions appeared for the first time at the early fifties, in a seminal paper by John Nash. Soon other mathematicians were interested on them, and some relevant applications found, but also soon their bad cohomological behaviour was realized. This put an end to a possible systematic development of the theory of Nash functions. Since that moment many specialists have devoted time and efforts to understand the nature of these functions, but only after fifty years we start to understand what is behind their failures, and how the problems generated by those failures should be settled. During this fifty years process the notion of Nash function has evolved in progressively more general settings, with an appealing feedback from concrete to abstract and viceversa. Moreover, a maze of surprising links have been revealed among all the questions involved. These are depicted in the Nash Labyrinth shown below, where each box contains a solved problem, each arrow is an implication, and the doubled boxes are the sources of the flow of arguments. We will try to explain the problems and their solutions in the following sections: §1. Brief history of the topic. §2. The notion of Nash function. §3. Global properties and cohomological failures. §4. Formulation of the problems. §5. Solutions in the compact case. §6. Solutions in the non-compact case. §7. Nash functions over arbitrary real closed fields. §8. Abstract Nash functions.

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Geometry of Subanalytic and Semialgebraic Sets

Cited by 140 publications

References 0 publications

Quadratic growth and critical point stability of semi-algebraic functions

Quadratic growth and critical point stability of semi-algebraic functions

Semi-algebraic functions have small subdifferentials

Global Problems on Nash Functions.

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