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1. CMB 2001 (vol 44 pp. 223)
Extending the Archimedean Positivstellensatz to the Non-Compact Case A generalization of Schm\"udgen's Positivstellensatz is given which holds
for any basic closed semialgebraic set in $\mathbb{R}^n$ (compact or not).
The proof is an extension of W\"ormann's proof.
Categories:12D15, 14P10, 44A60 |
2. CMB 1999 (vol 42 pp. 354)
A Real Holomorphy Ring without the SchmÃ¼dgen Property 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.
Categories:12D15, 14P10, 44A60 |
3. CMB 1997 (vol 40 pp. 81)
Une caractÃ©risation des corps satisfaisant le thÃ©orÃ¨me de l'axe principal Resum\'e. On caract\'erise les corps $K$ satisfaisant le th\'eor\`eme
de l'axe principal \`a l'aide de propri\'et\'es des formes
carac\-t\'erisation de ces m\^emes corps due \`a Waterhouse,
on retrouve \`a partir de l\`a, de fa\c{c}on \'el\'ementaire,
un r\'esultat de Becker selon lequel un pro-$2$-groupe qui se
r\'ealise comme groupe de Galois absolu d'un tel corps $K$ est
engendr\'e par des involutions.
ABSTRACT. We characterize general fields $K$, satisfying the
Principal Axis Theorem, by means of properties of trace forms of
the finite extensions of $K$. From this and Waterhouse's
characterization of the same fields, we rediscover, in quite an
elementary way, a result of Becker according to which a
pro-$2$-group which occurs as the absolute Galois group of such
a field $K$, is generated by
Categories:11E10, 12D15 |