Universität Konstanz Tom-Lukas Kriel Department of Mathematics and Statistics María López Quijorna
Summer Term 2017 Markus Schweighofer
Real Algebraic Geometry II – Exercise Sheet 10
Exercise 1 (4P) Let f
∈
R[
X]
be homogeneous and positive definite. Show that there existsk∈
Nsuch that(
∑ni=1Xi2)
kf∈
∑R[
X]
2.Hint:Try to imitate the proof of Example 8.2.15. Use the substitutionXi
7→ √
XiX21+...+Xn2
and calculate in the fieldR
(
X)
qX12+
...+
X2n .Exercise 2(4P) LetKbe a subfield ofRequipped with the induced order and let Abe a commutative ring containingK. Suppose T is a preorder or Archimedean semiring of A,K≥0
⊆
Tand M⊆
Ais an Archimedean T-module of A. SetS:
= {
x∈
Rn| ∀
f∈
M : f(
x) ≥
0}
.Suppose that there are a1, ...,am
∈
A and f1, ..., fm∈
T such that f=
∑mi=1fiai, f≥
0 onSandai(
x) >
0 for all x∈
Swith f(
x) =
0. Show that f∈
M.Exercise 3(4P) Let Abe a commutative ring. We say that a preorderTissaturatedif it is an intersection of prime cones.
(a) Show that a preorder T of A is saturated if and only if for all f
∈
A, m∈
N, s,t∈
Tthe equalitys f=
t+
f2m implies f∈
T.(b) Suppose now thatA
=
R[
X]
and thatTis finitely generated. Define S= {
x∈
Rn| ∀
t∈
T: t(
x) ≥
0}
.Show thatTis saturated if and only if every polynomial f
∈
R[
X]
fulfilling f≥
0 onSis contained inT.Exercise 4(4P) LetAbe a commutative ring containingQ. Show that an Archimedean preorderTof Ais saturated if and only ifTm :
= (
A\
m)
−2T is saturated inAm :
= (
A\
m)
−1A for every maximal idealm⊆
A.Exercise 5(4P) Suppose thatT is an Archimedean preorder ofR
[
X]
, S= {
x∈
Rn| ∀
g∈
T : g(
x) ≥
0}
and f
∈
T+ (
f2)
satisfies f≥
0 onS. Show that f∈
T.Hint:Write f
=
t+
h f2witht∈
T andh∈
R[
X]
. Work with I := (
t,f2)
. Exercise 6(6P) Decide if the preorder generated by(a) 1
−
X2 (b)−
1+
X2 inR[
X]
is saturated.Please submit until Tuesday, July 4, 2017, 9:55 in the box named RAG II near to the room F411.
