Difference between revisions of "Semi-algebraic set"
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''semi-analytic set.'' | ''semi-analytic set.'' | ||
− | A semi-algebraic set in | + | A semi-algebraic set in $ \mathbf R ^ {n} $( |
+ | or $ k ^ {n} $, | ||
+ | where $ k $ | ||
+ | is a real closed field) is a set that can be given by finitely many polynomial equalities and inequalities. More precisely, for $ g \in \mathbf R [ X _ {1} \dots X _ {n} ] $, | ||
+ | let $ U ( g)= \{ {x \in \mathbf R ^ {n} } : {g( x)> 0 } \} $. | ||
+ | Then $ E $ | ||
+ | is semi-algebraic if it belongs to the smallest Boolean ring of subsets of $ \mathbf R ^ {n} $ | ||
+ | containing all the $ U( g) $. | ||
A semi-analytic set is, by definition, a set in a real-analytic manifold which can be locally described by finitely many analytic equalities and inequalities. | A semi-analytic set is, by definition, a set in a real-analytic manifold which can be locally described by finitely many analytic equalities and inequalities. | ||
− | The Tarski–Seidenberg theorem asserts the existence of a decision procedure (cf. also [[Decidable set|Decidable set]]) for deciding the truth of any elementary sentence built up from finitely many polynomial inequalities | + | The Tarski–Seidenberg theorem asserts the existence of a decision procedure (cf. also [[Decidable set|Decidable set]]) for deciding the truth of any elementary sentence built up from finitely many polynomial inequalities $ g _ {i} ( x _ {1} \dots x _ {n} ) > 0 $, |
+ | the connectives "and" , "or" and "not" , and the quantifiers $ \exists x _ {j} $, | ||
+ | $ \forall x _ {k} $. | ||
+ | Two precise formulations are: 1) Let $ E \subset \mathbf R ^ {n} $ | ||
+ | be a semi-algebraic set and $ \pi : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n-} 1 $ | ||
+ | the projection onto the last $ n- 1 $ | ||
+ | coordinates. Then $ \pi ( E) $ | ||
+ | is semi-algebraic. 2) Let $ S ( x _ {1} \dots x _ {n} ; t _ {1} \dots t _ {m} ) $ | ||
+ | be a finite sentence built up from inequalities $ p _ {i} ( x _ {1} \dots x _ {n} ; t _ {1} \dots t _ {m} ) > 0 $ | ||
+ | and the connectives "and" , "or" and "not" (such a sentence is called a polynomial relation). Let $ Q _ {1} \dots Q _ {n} $ | ||
+ | be a series of quantifiers of the form $ \exists x _ {j} $ | ||
+ | or $ \forall x _ {k} $. | ||
+ | There is an algorithm for finding a polynomial relation $ T( t _ {1} \dots t _ {m} ) $ | ||
+ | such that | ||
− | + | $$ | |
+ | T( t _ {1} \dots t _ {m} ) \iff \ | ||
+ | Q _ {1} \dots Q _ {n} S( x _ {1} \dots x _ {n} ; t _ {1} \dots t _ {m} ). | ||
+ | $$ | ||
− | It follows from the Tarski–Seidenberg theorem that the image under a polynomial mapping | + | It follows from the Tarski–Seidenberg theorem that the image under a polynomial mapping $ \mathbf R ^ {n} \rightarrow \mathbf R ^ {m} $ |
+ | of a semi-algebraic set is semi-algebraic. Indeed, this is equivalent. | ||
The image of a semi-analytic set under an analytic mapping is not necessarily semi-analytic. A subanalytic set on a real-analytic manifold is, by definition, a set that is locally the image of a semi-analytic set under an analytic mapping. The points of a subanalytic set at which it is not semi-analytic form a subanalytic set, cf. [[#References|[a2]]]. | The image of a semi-analytic set under an analytic mapping is not necessarily semi-analytic. A subanalytic set on a real-analytic manifold is, by definition, a set that is locally the image of a semi-analytic set under an analytic mapping. The points of a subanalytic set at which it is not semi-analytic form a subanalytic set, cf. [[#References|[a2]]]. |
Revision as of 08:13, 6 June 2020
semi-analytic set.
A semi-algebraic set in $ \mathbf R ^ {n} $( or $ k ^ {n} $, where $ k $ is a real closed field) is a set that can be given by finitely many polynomial equalities and inequalities. More precisely, for $ g \in \mathbf R [ X _ {1} \dots X _ {n} ] $, let $ U ( g)= \{ {x \in \mathbf R ^ {n} } : {g( x)> 0 } \} $. Then $ E $ is semi-algebraic if it belongs to the smallest Boolean ring of subsets of $ \mathbf R ^ {n} $ containing all the $ U( g) $.
A semi-analytic set is, by definition, a set in a real-analytic manifold which can be locally described by finitely many analytic equalities and inequalities.
The Tarski–Seidenberg theorem asserts the existence of a decision procedure (cf. also Decidable set) for deciding the truth of any elementary sentence built up from finitely many polynomial inequalities $ g _ {i} ( x _ {1} \dots x _ {n} ) > 0 $, the connectives "and" , "or" and "not" , and the quantifiers $ \exists x _ {j} $, $ \forall x _ {k} $. Two precise formulations are: 1) Let $ E \subset \mathbf R ^ {n} $ be a semi-algebraic set and $ \pi : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n-} 1 $ the projection onto the last $ n- 1 $ coordinates. Then $ \pi ( E) $ is semi-algebraic. 2) Let $ S ( x _ {1} \dots x _ {n} ; t _ {1} \dots t _ {m} ) $ be a finite sentence built up from inequalities $ p _ {i} ( x _ {1} \dots x _ {n} ; t _ {1} \dots t _ {m} ) > 0 $ and the connectives "and" , "or" and "not" (such a sentence is called a polynomial relation). Let $ Q _ {1} \dots Q _ {n} $ be a series of quantifiers of the form $ \exists x _ {j} $ or $ \forall x _ {k} $. There is an algorithm for finding a polynomial relation $ T( t _ {1} \dots t _ {m} ) $ such that
$$ T( t _ {1} \dots t _ {m} ) \iff \ Q _ {1} \dots Q _ {n} S( x _ {1} \dots x _ {n} ; t _ {1} \dots t _ {m} ). $$
It follows from the Tarski–Seidenberg theorem that the image under a polynomial mapping $ \mathbf R ^ {n} \rightarrow \mathbf R ^ {m} $ of a semi-algebraic set is semi-algebraic. Indeed, this is equivalent.
The image of a semi-analytic set under an analytic mapping is not necessarily semi-analytic. A subanalytic set on a real-analytic manifold is, by definition, a set that is locally the image of a semi-analytic set under an analytic mapping. The points of a subanalytic set at which it is not semi-analytic form a subanalytic set, cf. [a2].
The closure of a semi-algebraic (respectively, semi-analytic or subanalytic) set is again semi-algebraic (respectively, semi-analytic or subanalytic).
The image of a semi-algebraic (respectively, subanalytic) set under an algebraic (respectively, analytic) mapping is a semi-algebraic (respectively, subanalytic) set.
Finally, a semi-algebraic (respectively, semi-analytic or subanalytic) subset of a smooth algebraic (respectively, analytic or analytic) variety admits a smooth stratification, whose strata are semi-algebraic (respectively, semi-analytic or subanalytic) (and smooth).
References
[a1] | H. Hironaka, "Stratification and flatness" P. Holm (ed.) , Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School , Sijthoff & Noordhoff (1977) pp. 199–266 MR0499286 Zbl 0424.32004 |
[a2] | W. Pawtucki, "Points de Nash des ensembles sous-analytiques" , Amer. Math. Soc. (1990) |
[a3] | G.W. Brumfiel, "Partially ordered rings and semi-algebraic geometry" , Cambridge Univ. Press (1979) MR0553280 Zbl 0415.13015 |
Semi-algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-algebraic_set&oldid=23974