# 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
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Semi-algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-algebraic_set&oldid=48654