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''semi-analytic set.''
 
''semi-analytic set.''
  
A semi-algebraic set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839501.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839502.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839503.png" /> is a real closed field) is a set that can be given by finitely many polynomial equalities and inequalities. More precisely, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839504.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839505.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839506.png" /> is semi-algebraic if it belongs to the smallest Boolean ring of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839507.png" /> containing all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839508.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s0839509.png" />, the connectives "and" , "or" and "not" , and the quantifiers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395011.png" />. Two precise formulations are: 1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395012.png" /> be a semi-algebraic set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395013.png" /> the projection onto the last <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395014.png" /> coordinates. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395015.png" /> is semi-algebraic. 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395016.png" /> be a finite sentence built up from inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395017.png" /> and the connectives "and" , "or" and "not" (such a sentence is called a polynomial relation). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395018.png" /> be a series of quantifiers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395019.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395020.png" />. There is an algorithm for finding a polynomial relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395021.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395022.png" /></td> </tr></table>
+
$$
 +
T( t _ {1} \dots t _ {m} )  \iff \
 +
Q _ {1} \dots Q _ {n} S( x _ {1} \dots x _ {n} ; t _ {1} \dots t _ {m} ).
 +
$$
  
It follows from the Tarski–Seidenberg theorem that the image under a polynomial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083950/s08395023.png" /> of a semi-algebraic set is semi-algebraic. Indeed, this is equivalent.
+
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
How to Cite This Entry:
Semi-algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-algebraic_set&oldid=23974