Semi-algebraic set

semi-analytic set.

A semi-algebraic set in (or , where is a real closed field) is a set that can be given by finitely many polynomial equalities and inequalities. More precisely, for , let . Then is semi-algebraic if it belongs to the smallest Boolean ring of subsets of containing all the .

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 , the connectives "and" , "or" and "not" , and the quantifiers , . Two precise formulations are: 1) Let be a semi-algebraic set and the projection onto the last coordinates. Then is semi-algebraic. 2) Let be a finite sentence built up from inequalities and the connectives "and" , "or" and "not" (such a sentence is called a polynomial relation). Let be a series of quantifiers of the form or . There is an algorithm for finding a polynomial relation such that

It follows from the Tarski–Seidenberg theorem that the image under a polynomial mapping 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
[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)