Real Algebraic Geometry With A View Toward Systems Control and Free Positivity

2014 ◽  
Vol 11 (2) ◽  
pp. 977-1045
Author(s):  
Didier Henrion ◽  
Salma Kuhlmann ◽  
Victor Vinnikov
Keyword(s):  
Algebraic Geometry ◽  
Real Algebraic Geometry ◽  
Free Positivity ◽  
Systems Control

scholarly journals TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 347-358 ◽  
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.

scholarly journals A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

2009 ◽  
Vol 52 (1) ◽  
pp. 39-52 ◽  
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.

Hilbert’s 17th Problem. Quadratic Forms in Real Algebraic Geometry

1998 ◽  
pp. 103-131
Author(s):  
Jacek Bochnak ◽  
Michel Coste ◽  
Marie-Françoise Roy
Keyword(s):  
Algebraic Geometry ◽  
Quadratic Forms ◽  
Real Algebraic Geometry ◽  
17Th Problem

Rigidity and moduli space in Real Algebraic Geometry

2006 ◽  
Vol 335 (4) ◽  
pp. 751-766 ◽  
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Algebraic Geometry ◽  
Moduli Space ◽  
Real Algebraic Geometry

Real Algebraic Geometry

2013 ◽  
Author(s):  
Vladimir I. Arnold
Keyword(s):  
Algebraic Geometry ◽  
Real Algebraic Geometry
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