Polar varieties and computation of one point in each connected component of a smooth real algebraic set

2003 ◽  
Author(s):  
Mohab Safey El Din ◽  
Éric Schost
Keyword(s):  
Connected Component ◽  
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 ∞ .

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.

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 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

On the structure of semialgebraic sets over p-adic fields

1988 ◽  
Vol 53 (4) ◽  
pp. 1138-1164 ◽  
Author(s):  
Philip Scowcroft ◽  
Lou van den Dries
Keyword(s):  
Boolean Operations ◽  
Semialgebraic Sets ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Open Set ◽  
Real Analytic ◽  
Selection Lemma ◽  
Image Position ◽  
Analytic Maps ◽  
Curve Selection

In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]:Let V ⊂ Rm be a real algebraic set, and let U ⊂ Rm be an open set defined by finitely many polynomial inequalities:Lemma 3.1. If U ∩ V contains points arbitrarily close to the origin (that is if 0 ∈ Closure (U ∩ V)) then there exists a real analytic curvewith p(0) = 0 and with p(t) ∈ U ∩ V for t > 0.Of course, this result will also apply to semialgebraic sets (finite unions of sets U ∩ V), and by Tarski's theorem such sets are exactly the sets obtained from real varieties by means of the finite Boolean operations and the projection maps Rn+1 → Rn. If, in this tiny extension of Milnor's result, we replace ‘R’ everywhere by ‘Qp’, we obtain a p-adic curve selection lemma, a version of which we will prove in this essay. Semialgebraic sets, in the p-adic context, may be defined just as they are over the reals: namely, as those sets obtained from p-adic varieties by means of the finite Boolean operations and the projection maps . Analytic maps are maps whose coordinate functions are given locally by convergent power series.

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