Tarski-Seidenberg theorem
Theorem (Tarski-Seidenberg).
The set of semialgebraic sets is closed under projection.
That is, if is a semialgebraic set, and if is the projection onto the first coordinates, then is also semialgebraic.
Łojasiewicz generalized this theorem further. For this we need a bit of notation.
Let .
Suppose is any ring of real valued functions on
.
Define to be the smallest
set of subsets of , which contain the sets
for all ,
and is closed under finite union, finite intersection and complement
.
Let denote the ring of polynomials in
with coefficients in .
Theorem (Tarski-Seidenberg-Łojasiewicz).
Suppose that , is such that . Then the projection of onto the first variables is in .
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 |