Real Algebraic Geometry

TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

Journal of Symbolic Logic ◽

10.1017/jsl.2016.45 ◽

2017 ◽

Vol 82 (1) ◽

pp. 347-358 ◽

Cited By ~ 4

Author(s):

PABLO CUBIDES KOVACSICS ◽

LUCK DARNIÈRE ◽

EVA LEENKNEGT

Keyword(s):

Algebraic Geometry ◽

Open Subset ◽

Cell Decomposition ◽

Real Algebraic Geometry ◽

Dimension Theory ◽

Definable Set ◽

Image Position ◽

Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

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Aspect graphs of bodies of revolution with algorithms of real algebraic geometry

Algorithms in Algebraic Geometry and Applications ◽

10.1007/978-3-0348-9104-2_17 ◽

1996 ◽

pp. 353-364 ◽

Cited By ~ 1

Author(s):

M.-F. Roy ◽

T. Van Effelterre

Keyword(s):

Algebraic Geometry ◽

Real Algebraic Geometry ◽

Bodies Of Revolution

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A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-005-4 ◽

2009 ◽

Vol 52 (1) ◽

pp. 39-52 ◽

Cited By ~ 8

Author(s):

Jakob Cimprič

Keyword(s):

Algebraic Geometry ◽

Representation Theory ◽

Representation Theorem ◽

Commutative Rings ◽

Real Algebraic Geometry ◽

New Approach ◽

Noncommutative Version ◽

Quadratic Modules ◽

Rings With Involution ◽

Associative Rings

AbstractWe present a new approach to noncommutative real algebraic geometry based on the representation theory of C*-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative C*-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

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