scholarly journals Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic Set

2014 ◽  
Vol 24 (3) ◽  
pp. 1313-1343 ◽  
Author(s):  
Aurélien Greuet ◽  
Mohab Safey El Din
Keyword(s):  
Polynomial Optimization ◽  
Probabilistic Algorithm ◽  
Algebraic Set ◽  
Real Algebraic Set

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

HERMITIAN COMPLEXITY OF REAL POLYNOMIAL IDEALS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250026 ◽  
Author(s):  
JOHN P. D'ANGELO ◽  
MIHAI PUTINAR
Keyword(s):  
Necessary Conditions ◽  
Complex Geometry ◽  
Polynomial Optimization ◽  
Symmetric Polynomial ◽  
Holomorphic Maps ◽  
Real Polynomial ◽  
Algebraic Subset ◽  
Algebraic Set ◽  
Holomorphic Embedding

We define the Hermitian complexity of a real polynomial ideal and of a real algebraic subset of Cn. This concept is aimed at determining precise necessary conditions for a Hermitian symmetric polynomial to agree with a Hermitian squared norm on an algebraic set. The latter topic has been a central theme in modern polynomial optimization and in complex geometry, specifically related to the holomorphic embedding of pseudoconvex domain into balls, or the classification of proper holomorphic maps between balls.

scholarly journals Bounding the number of connected components of a real algebraic set

1991 ◽  
Vol 6 (2) ◽  
pp. 191-209 ◽  
Author(s):  
Riccardo Benedetti ◽  
Francois Loeser ◽  
Jean Jacques Risler
Keyword(s):  
Connected Components ◽  
Algebraic Set ◽  
Real Algebraic Set

scholarly journals Algebraic surfaces determine analyticity of functions

2021 ◽  
Author(s):  
Jacek Bochnak ◽  
Wojciech Kucharz
Keyword(s):  
Algebraic Surface ◽  
Algebraic Curves ◽  
Algebraic Surfaces ◽  
Connected Component ◽  
Algebraic Set ◽  
Real Algebraic Set

AbstractLet $$f :X \rightarrow \mathbb {R}$$ f : X → R be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction $$f|_{S}$$ f | S is an analytic (resp. a Nash) function for every nonsingular algebraic surface $$S \subset X$$ S ⊂ X whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dim$$X \ge 2$$ X ≥ 2 and f is of class $$\mathcal {C}^{\infty }$$ C ∞ .

scholarly journals On the link of a stratum in a real algebraic set

Topology ◽  
1992 ◽  
Vol 31 (2) ◽  
pp. 323-336 ◽  
Author(s):  
Michel Coste ◽  
Krzysztof Kurdyka
Keyword(s):  
Algebraic Set ◽  
Real Algebraic Set
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