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 .
Suppose that , is such that . Then the projection of onto the first variables is in .
- 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
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