# Download Real algebraic and semi-algebraic sets by R Benedetti PDF

By R Benedetti

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5 Continuity o f roots 35 P r o p o s itio n h is a homeomorphism. P r o o f. It is clear from the above remark that it is sufficient to show that h is continuous. Note that the proposition is trivial for n = 1. Let us assume that n > 2. a:n] G C(n) | |x,| < sm (i C> = {(a 0, . . , a n- i ) € C ’l ||ai | < s (j = 1, . . , n - 1)}, = 1 , . . , n ) }, C'J = g(Cs). Note that 1) if s' > s, then C's, is a neighbourhood of C's in Cn; 2) Cs is a compact subset of C ^ , for Cs is equal to 7r(C5), where Cs is the compact subset of Cn given by Cs = 7r_ 1([i?

To summarize Pj = 0 on T if there exist i and j as above such that j° = j. Other­ wise P j(xyt) ^ 0 for every (x ,t) E I\ Claim 3’ is proved. C la im 3” (with the notation of 3’): Actually T = T*. 2. 1). P r o o f o f 3” . (x0,ti) E T, Assume that there exists (#o,*o) € r'\ r . with, for instance, t0 < ti. Let ti be such that The family of polynomials { PjtX0(T ) E R [r ]}j= i... 2, because in addition to the initial polynomials q\, . . , dcqT/dtc, so that the above {x 0} x [t0,^i] C T' since { { z 0} X family R} we also have all the partial derivatives is closed under differentiation.

F P}. P r o p o s itio n Let A be a subring of O stable under derivation. Let B C O be a Liouville extension of A. Then, if A admits a separating algorihm, so does B. P r o o f. Let / i , . . , f q £ B. By the definition of a Liouville extension, there are <7i , . . , gr £ B such that / i , . . , f q £ A[gu . . , gr\ C £ , and, for each / = 1 , . . , r, either g f 1, or g\, or g'tff1 is in A. As our proof will be constructive, we may merely assume that g is given with B = A[g] and either <7-1 , or or g'g1 is in A.