Real algebraic geometry and the 17th Hilbert problem

1980 ◽  
Vol 251 (3) ◽  
pp. 213-241 ◽  
Author(s):  
Jacek Bochnak ◽  
Gustave Efroymson
Keyword(s):  
Algebraic Geometry ◽  
Hilbert Problem ◽  
Real Algebraic Geometry ◽  
17Th Hilbert Problem

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.

ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

2012 ◽  
Vol 23 (04) ◽  
pp. 1250031 ◽  
Author(s):  
JOSÉ F. FERNANDO ◽  
J. M. GAMBOA
Keyword(s):  
Integral Domain ◽  
Hilbert Problem ◽  
Semialgebraic Sets ◽  
Semialgebraic Set ◽  
Irreducible Components ◽  
Real Geometry ◽  
Nash Manifold ◽  
Substitution Theorem ◽  
Satisfactory Theory ◽  
17Th Hilbert Problem

In this work we define a semialgebraic set S ⊂ ℝn to be irreducible if the noetherian ring [Formula: see text] of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring [Formula: see text]: Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

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.

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
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