Tarski-Seidenberg theorem

Theorem (Tarski-Seidenberg).

That is, if An×m is a semialgebraic set, and if π is the projection onto the first n coordinates, then π(A) is also semialgebraic.

Łojasiewicz generalized this theorem further. For this we need a bit of notation.

Let Un. Suppose 𝒜(U) is any ring of real valued functions on U. Define 𝒮(𝒜(U)) to be the smallest set of subsets of U, which contain the sets {xUf(x)>0} for all f𝒜(U), and is closed under finite union, finite intersectionMathworldPlanetmath and complementMathworldPlanetmath. Let 𝒜(U)[t] denote the ring of polynomials in tm with coefficients in 𝒜(U).

Theorem (Tarski-Seidenberg-Łojasiewicz).

Suppose that VU×RmRn+m, is such that VS(A(U)[t]). Then the projection of V onto the first n variables is in S(A(U)).


  • 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