Tarski-Seidenberg theorem
Theorem (Tarski-Seidenberg).
The set of semialgebraic sets^{} is closed under projection.
That is, if $A\subset {\mathbb{R}}^{n}\times {\mathbb{R}}^{m}$ is a semialgebraic set, and if $\pi $ is the projection onto the first $n$ coordinates, then $\pi (A)$ is also semialgebraic.
Łojasiewicz generalized this theorem further. For this we need a bit of notation.
Let $U\subset {\mathbb{R}}^{n}$. Suppose $\mathcal{A}(U)$ is any ring of real valued functions on $U$. Define $\mathcal{S}(\mathcal{A}(U))$ to be the smallest set of subsets of $U$, which contain the sets $\{x\in U\mid f(x)>0\}$ for all $f\in \mathcal{A}(U)$, and is closed under finite union, finite intersection^{} and complement^{}. Let $\mathcal{A}(U)[t]$ denote the ring of polynomials in $t\in {\mathbb{R}}^{m}$ with coefficients in $\mathcal{A}(U)$.
Theorem (Tarski-Seidenberg-Łojasiewicz).
Suppose that $V\mathrm{\subset}U\mathrm{\times}{\mathrm{R}}^{m}\mathrm{\subset}{\mathrm{R}}^{n\mathrm{+}m}$, is such that $V\mathrm{\in}\mathrm{S}\mathit{}\mathrm{(}\mathrm{A}\mathit{}\mathrm{(}U\mathrm{)}\mathit{}\mathrm{[}t\mathrm{]}\mathrm{)}$. Then the projection of $V$ onto the first $n$ variables is in $\mathrm{S}\mathit{}\mathrm{(}\mathrm{A}\mathit{}\mathrm{(}U\mathrm{)}\mathrm{)}$.
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 | Tarski-Seidenberg theorem |
---|---|
Canonical name | TarskiSeidenbergTheorem |
Date of creation | 2013-03-22 16:46:13 |
Last modified on | 2013-03-22 16:46:13 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 5 |
Author | jirka (4157) |
Entry type | Theorem |
Classification | msc 14P15 |
Classification | msc 14P10 |
Related topic | SemialgebraicSet |
Related topic | SubanalyticSet |
Defines | Tarski-Seidenberg-Łojasiewicz theorem |