S`(T) with coefficients inQ such that ∀T ∈R2 : &#34

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Fachbereich Mathematik und Statistik Prof. Dr. Salma Kuhlmann

Lothar Sebastian Krapp Simon Müller

WS 2018 / 2019

Real Algebraic Geometry I

Exercise Sheet 6 Semialgebraic sets

Exercise 21 (4 points)

Let (K, P) be an ordered field and letRbe the real closure of (K, P). Recall thatRcan be equipped with the order topology and K can be considered as a topological subspace of R.

(a) Suppose that (K, P) is Archimedean. Show thatK is dense in R, i.e. thatR is the topological closure ofK inR.

(b) Construct an ordered field which is not dense in its real closure.

(Hint: Take a suitable fieldQ and consider Q(x) with a suitable ordering.)

Let R be a real closed field and letS(T , X) be the system

T₂X₁²+T₁X₂²+T₁T₂X₁+ 1 = 0,

whereT = (T₁, T₂) andX= (X₁, X₂). Find systems of equalities and inequalitiesS₁(T), . . . , S_`(T) with coefficients inQ such that

∀T ∈R² :

"

∃X∈R² :S(T , X)⇐⇒

`

_

i=1

Si(T)

# .

Let R be a real closed field.

(a) Show that the semialgebraic sets in R are exactly the finite unions of points in R and open intervals with endpoints in R∪ {∞,−∞}.

(b) Let m∈Nand let A be a semialgebraic subset of R^m. Show that for some n∈N, there is an algebraic set B ⊆ R^m+n such that π(B) = A, where π : R^m+n → R^m is the projection map introduced in Lecture 11.

(Hint: Find a polynomial f ∈ R[t, x], where t = (t₁, . . . , t_m) and x = (x₁, . . . , x_n), such that A={t∈R^m| ∃x∈Rⁿ:f(t, x) = 0}.)

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Let exp :R→R,x7→e^x be the standard exponential function onR. (a) LetS ⊆Rbe infinite and g∈R[X, Y] such that for anyx∈S

g(x,exp(x)) = 0.

Show that g= 0.

(Hint: IfSis bounded, use the identity theorem of complex analysis: Letf andgbe holomorphic, i.e. differentiable, functions from C to C. Suppose that the set {x ∈ C | f(x) = g(x)} has a limit point inC. Then f and g coincide on C, i.e. f(x) =g(x) for anyx∈C.)

(b) Deduce that exp is not a semialgebraic map.

Please hand in your solutions by Thursday, 06 December 2018, 08:15h (postbox 16 in F4).

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