1. CMB 2007 (vol 50 pp. 105)
 Klep, Igor

On Valuations, Places and Graded Rings Associated to $*$Orderings
We study natural $*$valuations, $*$places and graded $*$rings
associated with $*$ordered rings.
We prove that the natural $*$valuation is always quasiOre and is
even quasicommutative (\emph{i.e.,} the corresponding graded $*$ring is
commutative), provided the ring contains an imaginary unit.
Furthermore, it is proved that the graded $*$ring is isomorphic
to a twisted semigroup algebra. Our results are applied to answer a question
of Cimpri\v c regarding $*$orderability of quantum
groups.
Keywords:$*$orderings, valuations, rings with involution Categories:14P10, 16S30, 16W10 

2. CMB 2003 (vol 46 pp. 575)
 Marshall, M.

Optimization of Polynomial Functions
This paper develops a refinement of Lasserre's algorithm for
optimizing a polynomial on a basic closed semialgebraic set via
semidefinite programming and addresses an open question concerning the
duality gap. It is shown that, under certain natural stability
assumptions, the problem of optimization on a basic closed set reduces
to the compact case.
Categories:14P10, 46L05, 90C22 

3. CMB 2003 (vol 46 pp. 400)
 Marshall, M.

Approximating Positive Polynomials Using Sums of Squares
The paper considers the relationship between positive polynomials,
sums of squares and the multidimensional moment problem in the
general context of basic closed semialgebraic sets in real $n$space.
The emphasis is on the noncompact case and on quadratic module
representations as opposed to quadratic preordering presentations.
The paper clarifies the relationship between known results on the
algebraic side and on the functionalanalytic side and extends these
results in a variety of ways.
Categories:14P10, 44A60 

4. CMB 2001 (vol 44 pp. 223)
5. CMB 1999 (vol 42 pp. 354)
 Marshall, Murray A.

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 $N+f, Nf \in T$ for some integer $N$,
then $f \in \R$. This is in sharp contrast with the
Schm\"udgenW\"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 
