Semi-algebraic set

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