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
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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 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