semialgebraic set
Definition.
Consider the $A\subset {\mathbb{R}}^{n}$, defined by real polynomials ${p}_{j\mathrm{\ell}}$, $j=1,\mathrm{\dots},k$, $\mathrm{\ell}=1,\mathrm{\dots},m$, and the relations^{} ${\u03f5}_{j\mathrm{\ell}}$ where ${\u03f5}_{j\mathrm{\ell}}$ is $>$, $=$, or $$.
$$A=\bigcup _{\mathrm{\ell}=1}^{m}\{x\in {\mathbb{R}}^{n}\mid {p}_{j\mathrm{\ell}}(x){\u03f5}_{j\mathrm{\ell}}0,j=1,\mathrm{\dots},k\}.$$ | (1) |
Sets of this form are said to be semialgebraic.
Similarly as algebraic subvarieties^{}, finite union and intersection^{} of semialgebraic sets is still a semialgebraic set. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski-Seidenberg theorem says that they are also closed under projection.
On a dense open subset of $A$, $A$ is (locally) a submanifold^{}, and hence we can easily define the dimension of $A$ to be the largest dimension at points at which $A$ is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension.
References
- 1 Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42. http://www.ams.org/mathscinet-getitem?mr=89k:32011MR 89k:32011
Title | semialgebraic set |
---|---|
Canonical name | SemialgebraicSet |
Date of creation | 2013-03-22 16:46:10 |
Last modified on | 2013-03-22 16:46:10 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 5 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 14P10 |
Related topic | TarskiSeidenbergTheorem |
Related topic | SubanalyticSet |
Defines | semialgebraic |
Defines | dimension of a semialgebraic set |