A Real Holomorphy Ring without the Schmüdgen Property
Printed: Sep 1999
A preordering $T$ is constructed in the polynomial ring $A = \R
[t_1,t_2, \dots]$ (countably many variables) with the following two
properties: (1)~~For each $f \in A$ there exists an integer $N$
such that $-N \le f(P) \le N$ holds for all $P \in \Sper_T(A)$.
(2)~~For all $f \in A$, if $N+f, N-f \in T$ for some integer $N$,
then $f \in \R$. This is in sharp contrast with the
Schm\"udgen-W\"ormann result that for any preordering $T$ in a
finitely generated $\R$-algebra $A$, if property~(1) holds, then
for any $f \in A$, $f > 0$ on $\Sper_T(A) \Rightarrow f \in T$.
Also, adjoining to $A$ the square roots of the generators of $T$
yields a larger ring $C$ with these same two properties but with
$\Sigma C^2$ (the set of sums of squares) as the preordering.
12D15 - Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx]
14P10 - Semialgebraic sets and related spaces
44A60 - Moment problems