(b) Determine for what A∈symn×n(R) the set {λA|λ∈R}is totally ordered by

Fachbereich Mathematik und Statistik Prof. Dr. Salma Kuhlmann

Lothar Sebastian Krapp Simon Müller

WS 2018 / 2019

Real Algebraic Geometry I

Exercise Sheet 1

Orderings and Dedekind cuts

Exercise 1 (4 points)

Let n ∈N and let sym_n×n(R) denote the set of all symmetric (n×n)-matrices over R. A matrix A∈sym_n×n(R) is called positive semi-definite (psd) ifv^tAv≥0 for any v∈Rⁿ. Consider the relation≤ on sym_n×n(R) given by

A≤B :⇐⇒B−A is psd.

(a) Show that for n≥2, the relation ≤is a partial order on sym_n×n(R) which is NOT total.

(b) Determine for what A∈sym_n×n(R) the set {λA|λ∈R}is totally ordered by ≤.

Exercise 2 (4 points)

(a) Let be the relation on R[x] defined as follows: For any p(x) = a_nxⁿ+. . .+a₁x +a₀ and q(x) =b_mx^m+. . .+b₁x +b₀,

p(x)q(x) :⇐⇒a_i≤b_i for all i∈N0,

where we setai = 0 fori > nandbi = 0 fori > m. Show thatdefines a partial order onR[x]

which is NOT total.

(b) We denote the set of formal power series in one variableby

R[[x]] :=

( _∞ X

i=0

a_ixⁱ

a_i ∈R )

.

Note thatR[x] is a subset ofR[[x]].

Let ≤be the total order on R[x] given in Lecture 1. Show that≤ can be extended to a total order on R[[x]], i.e. find a total order ≤⁰ on R[[x]] such that for anyp(x), q(x)∈R[x],

p(x)≤⁰ q(x)⇐⇒p(x)≤q(x).

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(c) Let R^N be the set of all real-valued sequences. Find a total order ≤ on R^N and an order- preserving bijection

ϕ:R^N,≤→ R[[x]],≤⁰,

i.e. a bijection ϕ from R^N to R[[x]] such that for any a, b ∈ R^N, if a ≤ b, then ϕ(a) ≤⁰ ϕ(b), where≤⁰ is the total order from (b).

(d) For a sequence a= (an)_n∈N∈R^N we define the support of aby supp(a) :={n∈N|an6= 0}.

Let F ⊆ R^N be the set of all sequences with finite support. Describe the totally ordered set (ϕ(F),≤⁰).

Exercise 3 (4 points)

Let (K,≤) be an ordered field and denote byK^× the multiplicative group K\ {0}. Prove that the following are equivalent:

(i) (K,≤) is Archimedean.

(ii) For any a, b∈K^× there existsn∈Nsuch that|a|< n|b|and |b|< n|a|.

(iii) K contains no infinitesimal positive element.

(iv) Zis coterminal inK.

Exercise 4 (4 points)

(a) Let (Γ,≤) be a totally ordered set. Show that (Γ,≤) is Dedekind complete if and only if (Γ,≤) has no free Dedekind cut.

(b) Let (K,≤) be a Dedekind complete ordered field. Show that (K,≤) is Archimedean.

Please hand in your solutions by Friday, 02 November 2018, 15:00h(postbox 16 in F4).

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