Semidefinite representation for convex hulls\\of real algebraic curves
Claus Scheiderer
We prove that the closed convex hull of any one-dimensional \sa\ subset of $\R^n$ has a semidefinite representation, meaning that it can be written as a linear projection of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve $C$ and a compact \sa\ subset $K$ of its $\R$-points, the preordering $\scrP(K)$ of all regular functions on $C$ that are nonnegative on $K$ is known to be finitely generated. We prove that $\scrP(K)$ is stable, which means that uniform degree bounds exist for representing elements of $\scrP(K)$. We also extend this last result to the case where $K$ is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence from our results we establish the Helton-Nie conjecture in dimension two: Every convex \sa\ subset of $\R^2$ has a semidefinite representation.
1208.3865.tex
\documentclass[a4paper,twoside,reqno]{amsart}
\usepackage{amssymb}
\usepackage[mathscr]{eucal}
\usepackage[all]{xy}
\usepackage{graphicx}

\usepackage{hyperref}

\theoremstyle{plain}
\newtheorem{prop}{Proposition}[section]
\newtheorem{thm}[prop]{Theorem}
\newtheorem{cor}[prop]{Corollary}
\newtheorem{lem}[prop]{Lemma}

\theoremstyle{definition}
\newtheorem{dfn}[prop]{Definition}
\newtheorem{rem}[prop]{Remark}
\newtheorem{rems}[prop]{Remarks}
\newtheorem{example}[prop]{Example}
\newtheorem{fact}[prop]{Fact}
\newtheorem{lab}[prop]{}

\newcommand{\A}{{\mathbb{A}}}
\newcommand{\C}{{\mathbb{C}}}
\newcommand{\N}{{\mathbb{N}}}
\newcommand{\R}{{\mathbb{R}}}

\newcommand{\m}{{\mathfrak{m}}}
\newcommand{\p}{{\mathfrak{p}}}

\newcommand{\scrF}{{\mathscr{F}}}
\newcommand{\scrP}{{\mathscr{P}}}

\newcommand{\x}{{\mathtt{x}}}

\DeclareMathOperator{\conv}{conv}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\Max}{Max}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Quot}{Quot}
\DeclareMathOperator{\spn}{span}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Sper}{Sper}
\DeclareMathOperator{\supp}{supp}

\newcommand{\Ex}{\mathrm{Ex}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\interior}{\mathrm{int}}
\newcommand{\rc}{\mathrm{rc}}

\newcommand{\isoto}{\overset{\sim}{\to}}
\newcommand{\labelto}[1]{\overset{#1}{\longrightarrow}}
\newcommand{\into}{\hookrightarrow}
\newcommand{\To}{\Rightarrow}
\newcommand{\Ot}{\Leftarrow}
\renewcommand{\iff}{\Leftrightarrow}
\newcommand{\wt}[1]{\widetilde{#1}}
\renewcommand{\subset}{\subseteq}
\renewcommand{\supset}{\supseteq}

\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\plus}{{\scriptscriptstyle+}}
\newcommand{\all}{\forall\>}
\newcommand{\ex}{\exists\>}
\newcommand\lal{\looparrowleft}

\renewcommand{\emptyset}{\varnothing}
\renewcommand{\setminus}{\smallsetminus}
\renewcommand{\epsilon}{\varepsilon}

\newcommand{\bil}[2]{\langle{#1},{#2}\rangle}
\newcommand{\comp}{\mathbin{\scriptstyle\circ}}

\newcommand{\ontopof}[2]{\genfrac{}{}{0pt}{}{#1}{#2}}
\renewcommand{\choose}[2]{\genfrac(){0pt}{}{#1}{#2}}

\newcommand{\sa}{semi-algebraic}

%===================================================================%

\begin{document}

\title
[Semidefinite representation for convex hulls of curves]
{Semidefinite representation for convex hulls\\of real algebraic
  curves}

\author{Claus Scheiderer}
\address
  {Fachbereich Mathematik and Statistik \\
  Universit\"at Konstanz \\
  D--78457 Konstanz \\
  Germany}
\email{claus.scheiderer@uni.constanz.de}
\urladdr{http://www.math.uni-konstanz/\textasciitilde scheider}

\begin{abstract}
We prove that the closed convex hull of any one-dimensional \sa\
subset of $\R^n$ has a semidefinite representation, meaning that it
can be written as a linear projection of the solution set of some
linear matrix inequality.
This is proved by an application of the moment relaxation method.
Given a nonsingular affine real algebraic curve $C$ and a compact
\sa\ subset $K$ of its $\R$-points, the preordering $\scrP(K)$ of all
regular functions on $C$ that are nonnegative on $K$ is known to be
finitely generated. We prove that $\scrP(K)$ is stable, which means
that uniform degree bounds exist for representing elements of
$\scrP(K)$. We also extend this last result to the case where $K$ is
only virtually compact.
The main technical tool for the proof of stability is the archimedean
local-global principle.
As a consequence from our results we establish the Helton-Nie
conjecture in dimension two: Every convex \sa\ subset of $\R^2$ has a
semidefinite representation.
\end{abstract}

\date\today
\maketitle

%-------------------------------------------------------------------%

\section*{Introduction}

Let $K\subset\R^n$ be a real algebraic or \sa\ set. The question of
how to represent the convex hull $\conv(K)$ of $K$ has attracted
growing attention in recent years. A good part of this interest
originates from optimization theory, namely from the problem of
optimizing a linear functional over $K$. One of the most promising
approaches that have been discussed is to express $\conv(K)$ (at
least up to taking closures) as a linear projection of a
spectrahedron, that is, of a set described by a linear matrix
inequality. In other words, one would like to find symmetric real
matrices $M_i$, $N_j$ of some size (for $0\le i\le n$, $1\le j\le k$
and some $k$) such that, writing
$$M(x,y)\>=\>M_0+\sum_{i=1}^nx_iM_i+\sum_{j=1}^ky_jN_j,$$
the closure of $\conv(K)$ coincides with the closure of the set
\begin{equation}\label{sdpset}%
S\>=\>\{x\in\R^n\colon\>\ex y\in\R^k\ M(x,y)\succeq0\}.
\end{equation}
(Here $M\succeq0$ means that the symmetric matrix $M$ is positive
semidefinite.) In view of the very efficient methods available in
semidefinite programming, such a representation is perfectly well
suited for optimizing linear functionals over~$K$.

Another approach tries to understand the set $\conv(K)$ via the dual
algebraic variety of the Zariski closure of its boundary, see
\cite{rs1} and \cite{rs2} for more details.

The question of characterizing the class of projected spectrahedra
was raised by Nemirovski in his plenary address at the ICM in Madrid
\cite{nm}.
Any projected spectrahedron is clearly a convex \sa\ set. No other
restriction is currently known. Helton and Nie have conjectured
\cite{hn1} that conversely every convex \sa\ set has a semidefinite
representation, by which one means, can be written as a projected
spectrahedron. A variety of sufficient conditions for the existence
of a semidefinite representation have been proved, see e.g.\
\cite{hn1}, \cite{hn2}, % Helton-Nie 2009/2010
\cite{ns}, % Netzer-Sinn
\cite{la}, % Lasserre 2009
\cite{nt}, % Netzer 2010
\cite{gpt}, % Gouveia-Parrilo-Thomas 2010
\cite{gn}, % Gouveia-Netzer
\cite{nipast}, % Nie-Parrilo-Sturmfels, k-ellipse, 2008
\cite{ni}, % Nie 2011?
\cite{sch:conv}, % Scheiderer 2011
and see also chapters 6 and~7 of the forthcoming book \cite{bpt} for
an overview of the current state of knowledge.
In this paper we prove
the existence of a semidefinite representation for the closed convex
hull of any one-dimensional \sa\ set. From this we derive a proof of
the Helton-Nie conjecture in dimension two.

For the construction of semidefinite representations we use the
moment relaxation method that was introduced by Lasserre and Parrilo
(\cite{la}, \cite{pa}, see also \cite{la:book}, \cite{bpt}).
Computing the convex hull of a set $K\subset\R^n$ (that we assume to
be basic closed \sa) means to determine the linear moments of all
probability measures on $K$ for which these moments exist.
By considering finite-dimensional relaxations of the $K$-moment
problem,
one obtains a nested hierarchy $K(1)\supset K(2)\supset\cdots$ of
explicit projected spectrahedra that contain $K$. Let us assume that
$K$ is compact. Then the sets $K(d)$ approximate $\conv(K)$
arbitrarily closely. When $K$ is a compact real algebraic set, the
approximating convex sets $TH(d)=\ol{K(d)}$ have been studied in the
literature under the name \emph{theta bodies} of $K$ (see \cite{gpt}
and \cite{bpt}, ch.~7). The approximation is known to become exact,
that is, $K(d)=\conv(K)$ holds for some number $d$, if and only if
every linear polynomial that is nonnegative on $K$ has a weighted sum
of squares representation with uniform degree bounds on the summands.
See Theorem \ref{relaxsumm} below for a rigorous formulation.

We consider a nonsingular affine algebraic curve $C$ over $\R$ and a
compact \sa\ subset $K$ of $C(\R)$, the set of real points on $C$. We
work in $\R[C]$, the affine coordinate ring of $C$. The saturated
preordering $\scrP(K)$ consists of all elements of $\R[C]$ that are
nonnegative on $K$. It is known \cite{sch:mz} that $\scrP(K)$ is
finitely generated. This means that there exist finitely many
elements $1=h_0,\,h_1,\dots,h_r\in\scrP(K)$ such that every $f\in
\scrP(K)$ has a representation
\begin{equation}\label{pkrep}%
f\>=\>\sum_{i=0}^r\sum_jp_{ij}^2h_i
\end{equation}
with $p_{ij}\in\R[C]$. Fixing $C$, $K$ and the $h_i$, the main result
of this paper (Corollary \ref{mainstab}) says that there exist
universal degree bounds for such representations. That is, every
$f\in\scrP(K)$ has some representation \eqref{pkrep} in which the
degrees of the summands are bounded above by some number that depends
only on $\deg(f)$.
(We are using degrees here to simplify the exposition, and so we
tacitly assume that $C$ is considered with a fixed embedding in some
affine space.) Technically, this result is expressed by saying that
the preordering $\scrP(K)$ is stable.
From this it follows that, for any morphism $\phi\colon C\to\A^n$
into affine space of any dimension, the relaxation process for the
convex hull of $\phi(K)$ in $\R^n$ becomes exact. In fact, this
latter property is equivalent to stability of $\scrP(K)$.

Our method for proving stability of $\scrP(K)$ may be of interest in
that we do not show the existence of degree bounds directly. Rather,
we establish the following equivalent fact: For any real closed field
$R$ containing $\R$, the preordering generated by the $h_i$ in
$R[C]=\R[C]\otimes R$ is again saturated (Theorem \ref{main}). This
fact, in turn, is
proved by an application of the archimedean local-global principle
\cite{sch:surf}, which allows us to work in local rings. Since the
field $R$ is non-archimedean, this seems impossible at first. We are
getting around this problem by working in the ring $B[C]=\R[C]\otimes
B$, rather than in $R[C]$, where $B$ is the smallest convex subring
of $R$ that contains~$\R$ (a non-noetherian valuation ring). We
believe that this way of applying the archimedean local-global
principle is novel and somewhat unexpected.

In the case where $C$ has genus one and $K=C(\R)$ is the full real
curve (assumed to be compact), our main result is known by
\cite{sch:conv}. In that paper, using arguments of Riemann-Roch type,
we had been able to give degree bounds of quite explicit nature,
resulting in bounds for the sizes of the derived exact semidefinite
representations.
For all curves of higher genus (as well as for genus one and $K\ne
C(\R)$), our present results should be new. In contrast to the method
from \cite{sch:conv}, the techniques developed in the present paper
do unfortunately not seem to give any explicit degree bounds.

From stability of $\scrP(K)$, for compact \sa\ sets $K$ on
nonsingular curves, we deduce the existence of a semidefinite
representation for the convex hull of any compact \sa\ set $S\subset
\R^n$ with $\dim(S)\le1$ (Theorem \ref{sdp}), using the moment
relaxation process. This case in turn implies the existence of such a
representation for the closed convex hull of any \sa\ set $S$ with
$\dim(S)\le1$, not necessarily compact (Theorem \ref{sdpclosed}).
From this we derive the Helton-Nie conjecture in dimension two
(Theorem \ref{hndim2}).

On the other hand, we extend the stability result to certain
noncompact cases. Namely, when $C$ is a nonsingular affine curve and
$K\subset C(\R)$ is a closed \sa\ set that is merely virtually
compact (see \ref{virtcptrappl}), the saturated preordering
$\scrP(K)$ is still finitely generated and stable (Theorem
\ref{vcptmain}). Again, this is proved by a reduction to the compact
case.

The paper is organized as follows. In Section~\ref{sect:relaxrappl}
we give a brief account of the relaxation method for constructing
semidefinite representations of convex hulls, in the generality that
is needed here. Section~\ref{sect:aux} contains auxiliary results for
working in the ring $\R[C]\otimes B$. This ring plays a key role in
the proof of stability of $\scrP(K)$ in the compact case
(Section~\ref{sect:main}). The existence of semidefinite
representations for compact convex hulls is deduced in
Section~\ref{sect:sdp}, whereas the extension to closed convex hulls
of arbitrary one-dimensional sets is contained in
Section~\ref{sect:noncpt}. Finally, Section~\ref{sect:virtcpt}
contains the proof of stability in the virtually compact case.

I would like to thank Tim Netzer for very useful remarks on a
preliminary version. His comments have led to substantial
improvements of some of the results.

%-------------------------------------------------------------------%

\section{Notations and preliminaries}

By an affine variety over a field $k$ we mean a reduced affine
$k$-scheme of finite type, that is, the Zariski spectrum $V=
\Spec(A)$ of a finitely generated reduced $k$-algebra~$A$. As usual,
we also write $A=k[V]$. If $E$ is any $k$-algebra, then $V(E)=\Hom_k
(A,E)$ denotes the set of $E$-valued points of $V$.

For the notion of real spectrum we refer to \cite{bcr}, \cite{pd},
\cite{ma} or \cite{sch:guide}. The real spectrum of a ring $A$ is
denoted by $\Sper(A)$. A convenient way to think of the points of
$\Sper(A)$ is that every such point is represented by a homomorphism
$A\to R$ into some real closed field $R$. Two homomorphisms $A\to
R_i$ ($i=1,2$) represent the same point of $\Sper(A)$ if, and only
if, they can be amalgamated over $A$, that is, if and only if there
exist $A$-embeddings $R_i\to R$ ($i=1,2$) into a common real closed
field~$R$.

For the notions of preordering and quadratic module in a ring $A$,
see one of \cite{pd}, \cite{ma} or \cite{sch:guide}. Recall that a
preordering $T$ in $A$ is called \emph{saturated} if it contains
every element of $A$ which is nonnegative in all points of $\Sper(A)$
where all elements of $T$ are nonnegative.

Let $R$ be a real closed field, and let $V$ be an affine $R$-variety.
Given a \sa\ subset $K$ of $V(R)$, we denote the associated
constructible subset of $\Sper R[V]$ by $\wt K$, as usual. The
saturated preordering associated with $K$ is denoted $\scrP(K)$, that
is,
$$\scrP(K)\>=\>\{f\in R[V]\colon f|_K\ge0\}.$$

The convex hull of a set $S\subset\R^n$ is denoted $\conv(S)$. The
set of extremal points of a convex subset $K\subset\R^n$ is denoted
$\Ex(K)$.

%-------------------------------------------------------------------%

\section{The relaxation method}\label{sect:relaxrappl}%

\begin{lab}
Let $A$ be a finitely generated $\R$-algebra, and let $M$ be a
finitely generated quadratic module in $A$, say $M=\Sigma_Ah_0+\cdots
+\Sigma_Ah_r$ with $1=h_0,\,h_1,\dots,h_r\in A$ and $\Sigma_A:=\Sigma
A^2$ (the cone of sums of squares in $A$). The quadratic module $M$
is said to be \emph{stable} (see \cite{ps}, \cite{sch:stab}) if,
given any finite-dimensional linear subspace $U$ of $A$, there exist
finite-dimensional linear subspaces $W_0,\dots,W_r$ of $A$ such that
$$M\cap U\>\subset\>\Sigma_{W_0}h_0+\cdots+\Sigma_{W_r}h_r,$$
where $\Sigma_{W_i}$ denotes the set of sums of squares of elements
of $W_i$. This property does not depend on the choice of the
generators $h_0,\dots,h_r$ of $M$. If $A$ is a polynomial ring over
$\R$, then stability of $M$ means that there exists a map $\varphi
\colon\N\to\N$ such that for every $f\in M$ there exists a
representation $f=\sum_{i,j}p_{ij}^2h_i$ with polynomials $p_{ij}$
and with $\deg(p_{ij}^2h_i)\le\varphi(\deg(f))$ for all $i,j$.
\end{lab}

\begin{lab}
By a \emph{semidefinite representation} of a (convex \sa) set
$S\subset\R^n$, one means a representation of $S$ in the form
\eqref{sdpset}, with suitable symmetric real matrices $M_i$, $N_j$.
Any set $S$ having such a representation is called a \emph{projected
spectrahedron}.
\end{lab}

We now recall the method of moment relaxation \cite{la} for
constructing semidefinite representations, in a generality adapted to
our needs. For more background we refer to Chapter~11 of
\cite{la:book}, and to Chapters 5 and~7 of the forthcoming book
\cite{bpt}. We only outline the basic principle of the construction,
ignoring possible refinements.

\begin{lab}\label{relaxrappl}%
Let $A$ be a finitely generated reduced $\R$-algebra. We denote the
associated affine $\R$-variety by $V=\Spec(A)$, and we always equip
the set $V(\R)=\Hom(A,\R)$ of real points of $V$ with its natural
Euclidean topology. Fix elements $1=h_0,\,h_1,\dots,h_r\in A$, write
$\Sigma_A:=\Sigma A^2$ for the cone of some of squares in $A$, let
$$M\>=\>\Sigma_Ah_0+\cdots+\Sigma_Ah_r$$
be the quadratic module in $A$ generated by the $h_i$, and let
$$K\>=\>\{\xi\in V(\R)\colon h_1(\xi)\ge0,\dots,h_r(\xi)\ge0\}$$
be the associated basic closed \sa\ subset of $V(\R)$. We assume that
$K$ is Zariski dense in $V$. Fix a finite-dimensional vector subspace
$L\subset A$ containing~$1$, and let $1,\,x_1,\dots,x_n$ be a basis
of $L$. We consider the morphism $\phi=\phi_L=(x_1,\dots,x_n)$ from
$V$ to affine $n$-space determined by $L$, and the induced map $\phi
\colon V(\R)\to\R^n$.

Fix a tuple $W=(W_0,\dots,W_r)$ of finite-dimensional linear
subspaces of $A$, and consider the linear subspace
$$U\>:=\>W_0W_0+h_1W_1W_1+\cdots+h_rW_rW_r$$
of $A$. We assume that $L$ is contained in $U$, and we denote by
$\rho\colon U'\to L'$ the restriction map between the dual linear
spaces. By $U'_1$ (resp.\ $L'_1$) we denote the set of all linear
forms $\lambda$ in $U'$ (resp.\ in $L'$) with $\lambda(1)=1$. We
identify $\R^n$ with $L'_1$ via the map
$$L'_1\>\labelto\sim\>\R^n,\quad\lambda\>\mapsto\>\bigl(\lambda
(x_1),\dots,\lambda(x_n)\bigr).$$
For $i=0,\dots,r$ let $\Sigma_{W_i}\subset W_iW_i$ denote the cone of
sums of squares of elements of $W_i$. The set
$$M_W\>:=\Sigma_{W_0}+h_1\Sigma_{W_1}+\cdots+h_r\Sigma_{W_r}$$
is contained in $M\cap U$ and is a convex \sa\ cone in $U$. Since $K$
is Zariski dense in $V$, we have $M\cap(-M)=\{0\}$. This implies that
$M_W$ is closed in $U$ (\cite{ps} Prop.\ 2.6).
Let $M_W^*\subset U'$ be the dual cone of $M_W$. Then $M_W^*$ can
be defined by a (homogeneous) linear matrix inequality, that is,
$M_W^*$ is a spectrahedral cone in $U'$. The subset $M_W^*\cap U'_1$
of $M_W^*$ is therefore a spectrahedron as well. Its image set
$$K_W\>:=\>\rho(M_W^*\cap U'_1)\>=\>L'_1\cap\rho(M_W^*)\>\subset\>
\R^n$$
under the restriction map $\rho\colon U'_1\to L'_1=\R^n$ is therefore
a projected spectrahedron in $\R^n$. For every $\xi\in K$, the cone
$M_W^*$ obviously contains the evaluation map at $\xi$ (restricted to
$U$). Therefore $K_W$ contains the set $\phi(K)$, and hence we also
have $\conv(\phi(K))\subset K_W$. Increasing the subspaces $W_0,
\dots,W_r$ of $A$ results in making $K_W$ smaller. The main facts are
summarized in the following theorem (c.f.\ \cite{la} Theorem~2):
\end{lab}

\begin{thm}\label{relaxsumm}%
\hfil
\begin{itemize}
\item[(a)]
$\ol{K_W}\>=\>\{\eta\in\R^n\colon\all f\in L\cap M_W$ $f(\eta)
\ge0\}$;
\item[(b)]
the inclusion $\ol{\conv(\phi(K))}\subset\ol{K_W}$ of closed convex
sets is an equality if and only if $L\cap\scrP(K)\subset M_W$;
\item[(c)]
if $M$ is archimedean then $\conv(\phi(K))=\bigcap_WK_W$,
intersection over all systems $W=(W_0,\dots,W_r)$ of
finite-dimensional subspaces of $A$.
\end{itemize}
\end{thm}

If $K$ is compact then $\conv(\phi(K))$ is again compact by
Carath\'eodory's lemma, and we get:

\begin{cor}\label{relaxcpt}%
If $K$ is compact, then $\conv(\phi(K))=K_W$ holds if and only if
$L\cap\scrP(K)\subset M_W$.
\end{cor}

\begin{lab}
The moment relaxation is said to become \emph{exact} when equality
$\ol{\conv(\phi(K))}=\ol{K_W}$ holds for some choice of
finite-dimensional subspaces $W=(W_0,\dots,W_r)$. When $K$ is
compact, this is equivalent to $\conv(\phi(K))=K_W$.

If one is aiming at describing the convex hull of $\phi(K)$ in
$\R^n$ (approximately or exactly), note that there is a two-fold
freedom of modifying the above construction. On the one hand, we may
enlarge the subspaces $W_0,\dots,W_r$. We may as well enlarge the
quadratic module $M$ by adding (finitely many) more generators $h_i$
from $\scrP(K)$. Both steps result in making the approximation
tighter. When the saturated preordering $\scrP(K)$ is itself finitely
generated, then choosing $M=\scrP(K)$ will give the best
approximations for $\conv(\phi(K))$.

When $K=V(\R)$ is a real algebraic set, and when an embedding
$V\subset\A^n$ has been fixed, the closed convex sets $\ol{K_W}
\subset\R^n$ resulting from $M=\Sigma\R[V]^2$ approximate the closed
convex hull $\ol{\conv(V(\R))}$. They have been studied under the
name \emph{theta bodies} by Gouveia, Parrilo, Thomas and others (see
\cite{gpt} and \cite{bpt}, Chapter~7).

Varying the embedding $\phi$, we see:
\end{lab}

\begin{cor}\label{allembiffstabl}%
Let $V$ be an affine $\R$-variety, let $K\subset V(\R)$ be a basic
closed set, Zariski dense in $V$, and assume that the saturated
preordering $\scrP(K)$ in $\R[V]$ is finitely generated. Then the
following two conditions are equivalent:
\begin{itemize}
\item[(i)]
For any $n\in\N$ and any morphism $\phi\colon V\to\A^n$ of
$\R$-varieties, the moment relaxation for the closed convex hull
$\ol{\conv(\phi(K))}$ becomes exact;
\item[(ii)]
the preordering $\scrP(K)$ in $\R[V]$ is stable.
\end{itemize}
\end{cor}

\begin{proof}
After fixing a finite description $\scrP(K)=\Sigma h_0+\cdots+\Sigma
h_r$ (with $\Sigma=\Sigma\R[V]^2$), stability of $\scrP(K)$ means
that for every finite-dimensional subspace $L\subset\R[V]$ there
exists a tuple $W=(W_0,\dots,W_r)$ of finite-dimensional subspaces
such that $L\cap\scrP(K)\subset M_W$. According to
\ref{relaxsumm}(b), this is equivalent to~(i).
\end{proof}

%-------------------------------------------------------------------%

\section{Auxiliary results}\label{sect:aux}%

Let $C$ be a nonsingular curve over $\R$. Here we collect results
that are needed for working in the base extension of $C$ to a real
closed valuation ring $B\supset\R$. The situation has some
resemblance with arithmetic surfaces. The main tool that will be
needed in the next section is Proposition \ref{nstordspez}.

\begin{lab}\label{wsorit}%
The following situation will be fixed. Let $R$ be a real closed field
containing $\R$, the field of real numbers, and let $B\subset R$ be
the convex hull of $\R$ in $R$. Then $B$ is a valuation ring of $R$,
and we denote by $v\colon R\to\Gamma\cup\{\infty\}$ the associated
Krull valuation. The maximal ideal of $B$ will be denoted by $\m$.
The residue field is $B/\m=\R$.

Let $A$ be a finitely generated $\R$-algebra, write $A_B=A\otimes B$
and $A_R=A\otimes R$ (with $\otimes:=\otimes_\R$ always). Given $0\ne
f\in A_R$ we can write $f=\sum_{i=1}^ra_i\otimes b_i$ with $a_i\in A$
and $b_i\in R$ in such a way that $a_1,\dots,a_r$ are linearly
independent over $\R$. Putting
$$w(f)\>:=\>\min\{v(b_i)\colon i=1,\dots,r\}$$
and $w(0):=\infty$ gives a well-defined map $w\colon A_R\to\Gamma\cup
\{\infty\}$ which extends the valuation~$v$.
For $f,\,g\in A_R$ we clearly have
$$w(f+g)\>\ge\>\min\{w(f),\,w(g)\}$$
and
$$w(fg)\>\ge\>w(f)+w(g),$$
and for $b\in R$ we have $w(bf)=w(f)+v(b)$.

The residue map $\pi\colon B\to B/\m=\R$ will often be denoted
$b\mapsto\ol b$. Accordingly we often denote the induced homomorphism
$A_B\to A$ by $f\mapsto\ol f$. We have $A_B=\{f\in A_R\colon w(f)\ge
0\}$, and for $f\in A_B$ we have $\ol f=0$ iff $w(f)>0$.
\end{lab}

\begin{lem}\label{geomintbew}%
Assume that the $\R$-algebra $A$ is integral. Then $w(fg)=w(f)+w(g)$
holds for all $f$, $g\in A_R$, and so $w$ extends to a valuation of
the field of fractions of $A_R$.
\end{lem}

Clearly, the residue field of the valuation $w$ of $\Quot(A_R)$ is
$\Quot(A)$.

\begin{proof}
$A$ integral implies that $A_R$ is integral, too. We can write $f=
af_0$ and $g=bg_0$, where $a$, $b\in R$ and $f_0$, $g_0\in A_B$
satisfy $w(f_0)=w(g_0)=0$. So we can assume
$w(f)=w(g)=0$, which means $\ol f$, $\ol g\ne0$. Since $A$ is a
domain we have $\ol f\cdot\ol g\ne0$, which implies $w(fg)=0$. The
lemma is proved.
\end{proof}

\begin{lab}\label{krprep}%
Write $V=\Spec(A)$ for the affine $\R$-scheme associated with $A$. We
need to work with the real spectrum of $A_B=A\otimes B$. Given any
point $\xi\in V(\C)=\Hom_\R(A,\C)$, we'll denote by $M_\xi$ the
kernel of the homomorphism $\xi\otimes\pi\colon A\otimes B\to\C$. So
$$M_\xi\>:=\>\Bigl\{\sum_ia_i\otimes b_i\in A\otimes B\colon\>\sum_i
a_i(\xi)\ol b_i=0\text{ in }\C\Bigr\}.$$
Clearly, $M_\xi$ is a maximal ideal of $A\otimes B$ whose residue
field is the residue field of $\xi$. When $\xi$ is real, i.e.\ $\xi
\in V(\R)$, there is a unique point in $\Sper(A\otimes B)$ whose
support is $M_\xi$. This point will be denoted $\alpha_\xi$.

We fix a \sa\ subset $K$ of $V(\R)$ and, as usual, denote by $\wt K$
the constructible subset of $\Sper(A)$ corresponding to~$K$. The
natural homomorphism $i\colon A\to A_B$ induces a continuous map
$i^*\colon\Sper(A_B)\to\Sper(A)$ of the real spectra, and we write
$X_K:=(i^*)^{-1}(\wt K)$. So $X_K$ is a constructible subset of
$\Sper(A_B)$, which is closed if $K$ is closed in $V(\R)$. By $K_R$
we denote the \sa\ subset of $V(R)$ that is defined by the same
inequalities as $K$. Considering $V(R)$ as a subset of $\Sper(A_B)$
in the natural way, we have $K_R=V(R)\cap X_K$.
\end{lab}

\begin{prop}\label{remaxidrelat}%
Assume that the \sa\ set $K$ is compact. Then the closed points of
$X_K$ are precisely the points $\alpha_\xi$, for $\xi\in K$.
\end{prop}

\begin{proof}
For $\xi\in K$ we have $\alpha_\xi\in X_K$ by construction, and this
is a closed point of $\Sper(A_B)$ since $\supp(\alpha_\xi)=M_\xi$ is
a maximal ideal of $A_B$.
Conversely, let $\alpha\in X_K$ be a point that is represented by a
homomorphism $\phi\colon A\otimes B\to S$ into a real closed field
$S$. Let $C\subset S$ be the convex hull of $\R$ in $S$, so we have
$C/\m_C=\R$. We claim that $\im(\phi)\subset C$ holds. Indeed, let
$a\in A$ and $b\in B$. Since $K$ is compact there is $c\in\R$ with
$|a|<c$ on $K$, and it follows that $|\phi(a\otimes1)|<c$ in $S$.
On the other hand, there is a real number $c'>0$ such that $|b|<c'$
holds on $\Sper(B)$, for example $c'=1+|\ol b|$. So we get $|\phi
(a\otimes b)|<cc'$ in $S$, whence $\phi(a\otimes b)\in C$. We can
therefore compose $\phi\colon A\otimes B\to C$ with the residue
homomorphism $C\to\R$, resulting in a homomorphism $\psi\colon
A\otimes B\to\R$. The point $\beta\in\Sper(A\otimes B)$ represented
by $\psi$ is a specialization of $\alpha$. On the other hand, $\beta=
\alpha_\xi$ for some $\xi\in V(\R)$, and the point $\xi$ lies in $K$.
\end{proof}

\begin{lem}\label{psdinox}%
Let $K\subset V(\R)$ be a \sa\ set, and let $f\in A_B$. Then $f$ is
nonnegative on the subset $X_K$ of $\Sper(A\otimes B)$ if, and only
if, $f$ is nonnegative on $K_R\subset V(R)$.
\end{lem}

\begin{proof}
The ``only if'' part is clear since $K_R\subset X_K$, see
\ref{krprep}. Conversely assume $f\ge0$ on $K_R$, and let $\alpha\in
X_K$. There exist a convex subring $W$ of $R$ that contains $B$, a
real closed field $E$ containing $\kappa_W=W/\m_W$ and a point
$\xi\in K_E\subset V(E)$ such that $\alpha$ is represented by the
homomorphism $A\otimes B\to E$, $a\otimes b\mapsto a(\xi)\ol b$.

There exists a homomorphic section of the residue map $W\to\kappa_W$.
Let $s\colon\kappa_W\to R$ be the field embedding induced by such a
section. After enlarging $E$ if necessary, there exists a
$\kappa_W$-embedding $\psi\colon R\into E$:
$$\begin{xy}
\xymatrix{%
\kappa_W \ar[r] \ar[d]_s & E \\
R \ar@{..>}[ur]_\psi}
\end{xy}$$
The homomorphism $A\otimes R\to E$, $a\otimes b\mapsto a(\xi)\cdot
\psi(b)$ represents a point $\beta\in\Sper(A_R)$ that lies in
$\wt{K_R}$. Let $\beta_0\in\Sper(A_B)$ be the image of $\beta$ under
the inclusion $\Sper(A_R)\subset\Sper(A_B)$. Then $\beta_0$ lies in
$X_K$ and specializes to $\alpha$.
Since $f\ge0$ on $K_R$ by assumption, we have $f\ge0$ in $\beta$
resp.\ in $\beta_0$, and therefore also  $f\ge0$ in $\alpha$.
\end{proof}

\begin{lab}\label{spezreeldisk}%
Now we specialize to the case where $C$ is an integral affine curve
over $\R$, and $A=\R[C]$ is the affine coordinate ring of $C$. We
keep fixing the extension $\R\subset R$ of real closed fields and the
convex hull $B$ of $\R$ in $R$, and we'll write $R[C]:=A\otimes R$
and $B[C]:=A\otimes B$. The following lemma is specific to the curves
case.
\end{lab}

\begin{lem}\label{spezreelrelat}%
Let $C$ be an integral affine curve over $\R$, and let $K\subset
C(\R)$ be a compact \sa\ set. Let $M$ be a maximal ideal of $\R[C]
\otimes B$, and assume that there exists $\alpha\in X_K$ with $\supp
(\alpha)\subset M$ and with $\supp(\alpha)\not\subset\R[C]\otimes\m$.
Then $M=M_\xi$ for some $\xi\in K$.
\end{lem}

\begin{proof}
Write $A=\R[C]$ as before. Let $P=\supp(\alpha)$ and write $\p=P\cap
A$. From $P\not\subset A\otimes\m$ it follows that $\p\ne(0)$, and
therefore $\p$ is a maximal ideal of $A$.
Since $P$ supports a point in $X_K$ we see that $\p$ is the maximal
ideal of $A$ corresponding to some point $\xi\in K$. Since $P':=
\p\otimes B$ is contained in $P$, and since $(A\otimes B)/P'=B$, we
conclude that necessarily $M=M_\xi$.
\end{proof}


\begin{lab}
We keep fixing the extension $\R\subset R$ and the valuation ring $B$
of $R$ as before. We assume now that $C$ is a nonsingular and
geometrically connected affine algebraic curve over $\R$, and we
consider the affine scheme $C\otimes_\R B=\Spec(\R[C]\otimes B)$.
This is a relative affine curve over $\Spec(B)$. If $B$ were a
discrete valuation ring, the situation would be a (very particular)
instance of a relative curve over a Dedekind scheme, hence an
arithmetic surface. However, $B$ has divisible value group and
therefore is not noetherian. Moreover, the Krull dimension of $B$ can
be arbitrarily large. Therefore we cannot directly rely on arguments
that are well-known for arithmetic surfaces, or simply for noetherian
rings. Still, the situation and the auxiliary results we are about to
prove, resemble the case of a relative curve over a discrete
valuation ring.
\end{lab}

\begin{lab}\label{spezdfn}%
Let $R'=R(\sqrt{-1})$ be the algebraic closure of $R$, and let $B'=
B[\sqrt{-1}]$, a valuation ring of $R'$ that extends the valuation
ring $B$ of $R$. The maximal ideal of $B'$ will be denoted $\m'$, and
we have $B'/\m'=\C$.
The valuation $v$ on $R$ (see \ref{wsorit}), resp.\ $w$ on $R(C)$
(see \ref{geomintbew}), extends uniquely to a valuation on $R'$,
resp.\ on $R'(C)$, and we use the same letter $v$, resp.\ $w$, to
denote this extension.
The residue field of the valuation $v$ on $R'$ is $\C$, and the
residue field of the valuation $w$ on $R'(C)$ is $\C(C)$, the complex
function field of the curve $C$. Given $g\in R'(C)$ with $w(g)\ge0$,
we denote the residue class of $g$ in $\C(C)$ by $\ol g$. Also, we
write $B'[C]=\R[C]\otimes B'$ and $R'[C]=\R[C]\otimes R'$. Again we
have $B'[C]=\{f\in R'[C]\colon w(f)=0\}$.

We consider the natural specialization map
$$C(B')\>\to\>C(\C),\quad\eta\mapsto\ol\eta$$
defined by composing homomorphisms $\R[C]\to B'$ with the residue
map $B'\to B'/\m'=\C$. Note that $\eta\in C(B')$ specializes to
$\xi\in C(\C)$ (that is, $\ol\eta=\xi$), if and only if $h(\eta)=0$
implies $\ol h(\xi)=0$, for every $h\in B[C]$.
Given $\xi\in C(\C)$, we'll use the notation
$$U(\xi)\>:=\>\{\eta\in C(B')\colon\ol\eta=\xi\}.$$
The maximal ideal of $B[C]$ associated with $\xi$ is denoted $M_\xi=
\{f\in B[C]\colon\ol f(\xi)=0\}$, see \ref{krprep}.

The zero or pole order of a rational function $g$ on a nonsingular
curve in a geometric point $\xi$ will be denoted by $\ord_\xi(g)$.
Thus, given $f\in B'[C]$ and $\eta\in C(R')$, the symbol $\ord_\eta
(f)$ denotes the vanishing order of $f$ in the point $\eta$ of the
generic fibre. For $\xi\in C(\C)$, on the other hand, the symbol
$\ord_\xi(\ol f)$ denotes the vanishing order in $\xi$ of the
restriction $\ol f$ of $f$ to the special fibre. Below (Proposition
\ref{nstordspez}) we'll show how the vanishing orders of $f$ in
points of the generic fibre determine the vanishing orders of $\ol f$
in the points of the special fibre.
\end{lab}

\begin{lem}\label{ord0quotbewr}%
Let $g\in R(C)^*$ satisfy $w(g)=0$, let $\xi\in C(\C)$ be a geometric
point of the special fibre, and assume $\ord_\eta(g)\ge0$ for every
$\eta\in U(\xi)$. Then there exist $0\ne f,\,h\in B[C]$ with
$g=\frac fh$ and $\ol h(\xi)\ne0$.
\end{lem}

In other words, $g$ lies in the localized ring $B[C]_{M_\xi}$.

\begin{proof}
We can write $g=\frac ba$ with $0\ne a,\,b\in R[C]$, and clearly we
can assume $w(a)=w(b)=0$.
This means that $a,\,b\in B[C]$.

Let $\eta_1,\dots,\eta_r$ be the zeros of $a$ in $U(\xi)$, and let
$\zeta_1,\dots,\zeta_s$ be the remaining zeros of $a$ in $C(R')$.
Since none of the $\zeta_j$ specializes to $\xi$, there exists $h\in
B[C]$ satisfying $\ol h(\xi)\ne0$ and $\ord_{\zeta_j}(h)\ge
\ord_{\zeta_j}(a)$ for $j=1,\dots,s$.

For any point $\eta\in C(R')$ we claim that $\ord_\eta(bh)\ge
\ord_\eta(a)$ holds. Indeed, this is trivial if $a(\eta)\ne0$. For
$\eta\in\{\zeta_1,\dots,\zeta_s\}$ it is so by the choice of $h$.
For $\eta\in\{\eta_1,\dots,\eta_r\}$ we have $\ord_\eta(b)\ge
\ord_\eta(a)$ by the assumption on $g$. So $gh=\frac{bh}a$ lies in
$R[C]$. Since $w(gh)=0$, we have $gh\in B[C]$.
\end{proof}

\begin{lem}\label{planecurvcase}%
Let $f,\,g\in B'[x,y]$ be such that the coefficient-wise reduced
polynomials $\ol f,\,\ol g\in\C[x,y]$ are not identically zero.
Assume $f(0,0)=g(0,0)=0$, and assume that the curves $\ol f=0$ and
$\ol g=0$ in $\C^2$ intersect transversally at $(0,0)$. Then the
curves $f=0$ and $g=0$ in $R'^2$ intersect transversally at $(0,0)$,
and they do not intersect in any point $(a,b)\ne(0,0)$ with $a,\,b\in
\m'$.
\end{lem}

\begin{proof}
The gradient vectors of $f$ and $g$ at the origin lie in $B'^2$, and
by assumption they are linearly independent modulo $\m'$. Hence they
are linearly independent in $R'^2$, which is the first assertion.
After a linear change of coordinates we can assume
$$f\>=\>x+\sum_{d\ge2}f_d(x,y),\quad g\>=\>y+\sum_{d\ge2}g_d(x,y)$$
where $f_d,\,g_d\in B'[x,y]$ are homogenous polynomials of degree
$d$, for $d\ge2$. Let $(0,0)\ne(a,b)\in\m'\times\m'$, and assume
$v(a)\le v(b)$. Since $v(a)>0$ we see that $v(f(a,b)-a)>v(a)$, whence
$v(f(a,b))=v(a)$, and therefore $f(a,b)\ne0$. Likewise, $v(a)\ge
v(b)$ implies $v(g(a,b))=v(b)$ and $g(a,b)\ne0$.
\end{proof}

\begin{lem}\label{spezordx2x}%
Let $\eta\in C(B')$, let $\xi=\ol\eta\in C(\C)$.
\begin{itemize}
\item[(a)]
There is $s\in B'[C]$ satisfying $s(\eta)=0$ and $\ord_{\xi}(\ol s)=
1$.
\item[(b)]
If $\eta\in C(B)$ then $s$ can be found in $B[C]$.
\item[(c)]
Any $s$ as in (a) satisfies $\ord_\eta(s)=1$ and $s(\eta')\ne0$ for
any $\eta'\in U(\xi)\setminus\{\eta\}$.
\end{itemize}
\end{lem}

\begin{proof}
Choose $t\in B'[C]$ such that $\ol t\in\C[C]$ is a local uniformizer
at $\ol\eta$. Then $t(\eta)\in\m'$. Putting $s:=t-t(\eta)$ gives
$s\in B'[C]$ with $s(\eta)=0$ and with $\ol s=\ol t$, hence
$\ord_{\ol\eta}(\ol s)=1$. If $\eta$ is real, i.e.\ $\eta\in C(B)$,
then $t$ (and therefore $s$) can be found in $B[C]$. This proves (a)
and (b).

(c)
The question is local around the point $\ol\eta\in C(\C)$. Zariski
locally around any given point, any nonsingular curve is isomorphic
to a Zariski open subset of a plane curve.
Therefore we can assume that $C$ is a (possibly singular) closed
curve in $\A^2_\C$, and that $\xi=(0,0)$ is a nonsingular point of
$C$. Now assertion (c) follows from Lemma \ref{planecurvcase}.
\end{proof}

\begin{prop}\label{nstordspez}%
Let $f\in B'[C]$ satisfy $w(f)=0$. The vanishing order of $\ol f$ in
a point $\xi\in C(\C)$ satisfies
$$\ord_\xi(\ol f)\>=\>\sum_{\eta\in U(\xi)}\ord_\eta(f).$$
\end{prop}

\begin{proof}
Let $e$ denote the right hand sum in the assertion, and let
$$\{\eta\in U(\xi)\colon\>f(\eta)=0\}\>=:\>\{\eta_1,\dots,\eta_r\}$$
(a finite set of points). For every $i=1,\dots,r$, choose $s_i\in
B'[C]$ with $w(s_i)=0$, $s_i(\eta_i)=0$ and $\ord_\xi(\ol s_i)=1$,
according to Lemma \ref{spezordx2x}(a). Moreover, put $e_i:=
\ord_{\eta_i}(f)$. Let $s:=s_1^{e_1}\cdots s_r^{e_r}\in B'[C]$, then
we have $w(s)=0$ and $\ord_\xi(\ol s)=e_1+\cdots+e_r=e$. For any
$\eta\in U(\xi)$ with $f(\eta)\ne0$ we have $s(\eta)\ne0$, by Lemma
\ref{spezordx2x}(c). Hence the rational function $g:=\frac fs\in
R'(C)^*$ has $\ord_\eta(g)=0$ for any $\eta\in C(B')$ with $\ol\eta=
\xi$. Applying Lemma \ref{ord0quotbewr} to $g$ and $g^{-1}$ shows
that $g$ is a unit in the localized ring $B'[C]_{M_\xi}$. Thus
$\ol g(\xi)\ne0$, and therefore $\ord_\xi(\ol f)=\ord_\xi(\ol s)=e$.
\end{proof}

%-------------------------------------------------------------------%

\section{Main theorem}\label{sect:main}%

The following fact is well-known:

\begin{thm}\label{satpofg}%
Let $C$ be a nonsingular affine curve over $\R$, and let $K\subset
C(\R)$ be a compact \sa\ set. Then the saturated preordering
$\scrP(K)$ of $K$ in $\R[C]$ is finitely generated.
\end{thm}

\begin{lab}\label{satpofgpf}%
See \cite{sch:mz} Theorem 5.21. More precisely, $\scrP(K)$ can be
generated by two elements (even as a quadratic module), and can in
fact be generated by a single element whenever $K$ has no isolated
points. If $K=C(\R)$ (assumed to be compact) we have $\scrP(K)=
\Sigma\R[C]^2$.

Let us briefly recall how this can be proved.
When $K$ has no isolated points, then $\scrP(K)$ is generated by any
$f\in\scrP(K)$ which has simple zeros in the boundary points of $K$
and has no other zeros in $K$ (one can show that such $f$ exists).
This follows from the archimedean local-global principle (see Theorem
\ref{archlgp} below).
In the general case, let $\xi_1,
\dots,\xi_r$ be the isolated points of $K$. We modify the set $K$ by
replacing each isolated point $\xi_i$ with a small closed interval
$[\xi_i,\eta_i]$ on $C(\R)$, for which $\eta_i\ne\xi_i$ lies on the
same connected component of $C(\R)$ as $\xi_i$, and the intervall is
so small that $[\xi_i,\eta_i]\cap K=\{\xi_i\}$. Let $K_1$ be the
modified set obtained in this way, and note that $K_1$ has no
isolated points. Let $K_2$ be a second such modification of $K$ in
which $\xi_i$ gets replaced by $[\eta_i',\xi_i]$, where $\eta'_i\ne
\xi_i$ is again chosen close to $\xi_i$, but such that $\eta_i$ and
$\eta'_i$ lie on opposite sides of $\xi_i$ on the local branch of
$C(\R)$ around $\xi_i$. Then, by the first part of the argument,
there exists a single generator $f_j$ of $\scrP(K_j)$, for both $j=1,
2$. Again using the archimedean local-global principle, one concludes
that $\scrP(K)$ is generated by $f_1$ and $f_2$.
\end{lab}

The following theorem, resp.\ its corollary (to which the theorem is
equivalent), is the first main result of this paper:

\begin{thm}\label{main}%
Let $C$ be a nonsingular affine curve over $\R$, let $K\subset C(\R)$
be a compact \sa\ set, and let $T=\scrP(K)$ be the saturated
preordering of $K$ in $\R[C]$. For any real closed field $R$
containing $\R$, the preordering $T_R$ generated by $T$ in $R[C]$ is
saturated as well.
\end{thm}

\begin{cor}\label{mainstab}%
For $C$ and $K$ as in \ref{main}, the preordering $\scrP(K)$ in
$\R[C]$ is stable.
\end{cor}

\begin{proof}
By \cite{sch:stab} Corollary 3.8, $T=\scrP(K)$ is stable if and only
if $T_R$ is saturated in $R[C]$, for every real closed field $R$
containing~$\R$. So the assertion follows from Theorem \ref{main}.
\end{proof}

\begin{rems}
\hfil
\smallskip

1.\
Assume that the nonsingular affine curve $C$ is irreducible. When $C$
is rational, the assertions of Theorem \ref{main} and Corollary
\ref{mainstab} are true regardless whether $K$ is compact or not.
More precisely, assume that $C$ is a nonsingular rational affine
curve, and let $K\subset C(\R)$ be any closed \sa\ subset. Then the
saturated preordering $\scrP(K)$ of $K$ in $\R[C]$ is finitely
generated, and is stable. This is well-known, and essentially
elementary.
\smallskip

2.\
When $C$ has genus one and $C(\R)$ is compact, \ref{main} and
\ref{mainstab} were proved for $K=C(\R)$ in \cite{sch:conv}. In all
other cases of positive genus, these results are new.
\smallskip

3.\
When $C$ has genus $\ge1$ and $K\subset C(\R)$ is a closed \sa\
set that is not compact, two situations can occur. Either $K$ is
virtually compact (see \ref{virtcptrappl} below); in this case we'll
prove later that the above results remain true (Theorem
\ref{vcptmain} below). Or else $K$ fails to be virtually compact;
then it is known that the preordering $\scrP(K)$ fails to be finitely
generated (\cite{sch:mz} Theorem 5.21), and so the notion of
stability does not even make sense for it.
\end{rems}

\begin{lab}
We start giving the proof of Theorem \ref{main}.
Let $B$ be the convex hull of $\R$ in $R$. We shall work in the ring
$B[C]=\R[C]\otimes B$, and shall use the auxiliary results from
Section~\ref{sect:aux}. In particular, we use the notation introduced
there. Let $T_B$ be the preordering generated by $T$ in $B[C]$.
Theorem \ref{main} asserts that $T_R$ contains every $g\in R[C]$ with
$g\ge0$ on $K_R$. It suffices to show that $T_B$ contains every $f\in
B[C]$ with $f\ge0$ on $K_R$ and with $w(f)=0$. Indeed, given $g\in
R[C]$ with $g\ge0$ on $K_R$, we find $0\ne b\in R$ with $w(g)=
v(b^2)$, and hence with $b^{-2}g\in B[C]$ and $w(b^{-2}g)=0$. Knowing
$b^{-2}g\in T_B$ clearly implies $g\in T_R$.
\end{lab}

The following easy observation is crucial for the entire proof:

\begin{lem}\label{archmodringerw}%
The preordering $T_B$ in $B[C]$ is archimedean.
\end{lem}

\begin{proof}
According to Schm\"udgen's theorem \cite{sm}, the preordering $T$ in
$\R[C]$ is archi\-medean. Let $f\in B[C]=\R[C]\otimes B$. Since $T-T
=\R[C]$, we can write $f$ in the form $f=\sum_{i=1}^rf_i\otimes b_i$
with $f_i\in T$ and $b_i\in B$ ($i=1,\dots,r$). Since $T$ is
archimedean, there exists $0<c_1\in\R$ with $c_1-f_i\in T$ ($i=1,
\dots,r$). By the definition of $B$ there exists $0<c_2\in R$ with
$b_i\le c_2$ in $R$ for every $i$, and hence $c_2-b_i$ is a square in
$B$ ($i=1,\dots,r$). We conclude that
$$rc_1c_2-f\>=\>c_2\sum_{i=1}^r(c_1-f_i)\otimes1+\sum_{i=1}^rf_i
\otimes(c_2-b_i)$$
lies in $T_B$.
\end{proof}

Lemma \ref{archmodringerw} allows us to use the archimedean
local-global principle, which we recall:

\begin{thm}\label{archlgp}%
\emph{(\cite{sch:surf} Corollary 2.10)}
Let $A$ be a ring (containing $\frac12$), let $P$ be an archimedean
preordering in $A$, and let $f$ be an element of the saturation of
$P$. Then $f$ lies in $P$ if (and only if) $f$ lies in $P_\m$ for
every maximal ideal $\m$ of $A$.
\end{thm}

Here $P_\m$ is the preordering generated by $P$ in the localized ring
$A_\m$.

\begin{lab}
Resuming the proof of Theorem \ref{main}, fix $f\in B[C]$ with $w(f)=
0$ and with $f\ge0$ on $K_R$. We have to show $f\in T_B$. From Lemma
\ref{psdinox} we know that $f\ge0$ on $X_K$, that is, $f$ lies in the
saturation of $T_B$ in $B[C]$.

According to Lemma \ref{archmodringerw}, we can apply Theorem
\ref{archlgp} to $T_B$. Therefore it suffices to prove, for every
maximal ideal $M$ of $B[C]$, that $f$ lies in $T_M$, the preordering
generated by $T$ in the local ring $B[C]_M$. Fix $M$, and let
$X_{K,M}=X_K\cap\Sper B[C]_M$. This is the basic closed constructible
subset of $\Sper B[C]_M$ associated with $T_M$.

If $f>0$ on $X_{K,M}$, then $f\in T_M$ by \cite{sch:lgp1},
Proposition 2.1. So we can assume that there exists $\alpha\in X_K$
with $f\in\supp(\alpha)\subset M$. The hypotheses of Lemma
\ref{spezreelrelat} apply to $M$, since $w(f)=0$ implies $\supp
(\alpha)\not\subset\R[C]\otimes\m$. By Lemma \ref{spezreelrelat},
therefore, we have $M=M_\xi$ for some point $\xi\in K$. From
\ref{spezdfn} recall the notation
$$U(\xi)\>=\>\{\eta\in C(B')\colon\>\ol\eta=\xi\}.$$
\end{lab}

\begin{lem}\label{chirurg}%
For every point $\eta\in U(\xi)$ with $f(\eta)=0$, there exists an
element $p_\eta\in T_B$ with $w(p_\eta)=0$ such that $p_\eta(\eta)=0$
and
$$\ord_\xi(\ol p_\eta)\>=\>\begin{cases}1&\text{if }\eta\in C(R),\
\eta\notin\interior(K_R),\\2&\text{if }\eta\in C(R),\ \eta\in
\interior(K_R),\\2&\text{if }\eta\notin C(R).\end{cases}$$
\end{lem}

Here $\interior(K_R)$ denotes the interior (relative to $C(R)$) of
the \sa\ set $K_R\subset C(R)$.

\begin{lab}\label{how2cmpl}%
Before giving the proof of Lemma \ref{chirurg}, we show how the
proof of Theorem \ref{main} can be completed using this lemma. Let
us decompose the set
$$\{\eta\in U(\xi)\cap C(R)\colon\>f(\eta)=0\}\>=\>\{\eta_1,\dots,
\eta_r\}\cup\{\zeta_1,\dots,\zeta_s\}$$
in such a way that the $\eta_i$ lie in $\interior(K_R)$ and the
$\zeta_j$ don't. Note that $f$ has even order in any of the points
$\eta_i$. Among the nonreal zeros of $f$ in $U(\xi)$, choose a
subset $\{\omega_1,\dots,\omega_t\}$ that contains exactly one
representative from each pair of complex conjugate points. Then put
$$p\>:=\>\prod_{i=1}^r(p_{\eta_i})^{\frac12\ord_{\eta_i}(f)}\cdot
\prod_{j=1}^s(p_{\zeta_j})^{\ord_{\zeta_j}(f)}\cdot\prod_{k=1}^t
(p_{\omega_k})^{\ord_{\omega_k}(f)},$$
with the $p_\eta$ chosen as in Lemma \ref{chirurg}.
Then $p\in T_B$. By Proposition \ref{nstordspez} we have
$$\ord_\xi(\ol f)\>=\>\sum_{i=1}^r\ord_{\eta_i}(f)+\sum_{j=1}^s\ord
_{\zeta_j}(f)+2\sum_{k=1}^t\ord_{\omega_k}(f),$$
and this number is also equal to $\ord_\xi(\ol p)$. It follows that
$g:=\frac fp$ is a unit in the local ring $B[C]_M$.

We would like $g$ to take nonnegative values in all real points $\eta
\in U(\xi)\cap K_R$. This obviously is the case whenever $p(\eta)\ne
0$. By continuity, it is also true whenever $\eta$ is not an isolated
point of $K_R$. The case when $\eta$ is an isolated point of $K_R$
can occur only for $\eta=\xi\in C(\R)$, and when $\xi$ is an isolated
point of $K$. This case can be dealt with by modifying one of the
local factors $p_\xi$ in the product $p$. We will explain this in the
course of the proof of Lemma \ref{chirurg}, see \ref{pfchirurg}.

Once this point is settled, the unit $g$ of $B[C]_M$ takes positive
values on the set $X_{K,M}$ associated with the preordering $T_M$,
using Lemma \ref{psdinox}. Hence, by another application of
\cite{sch:lgp1} Proposition 2.1, we conclude that $g$ lies in $T_M$.
As a consequence we get $f=pg\in T_M$, as desired.
\end{lab}

\begin{lab}\label{pfchirurg}%
For the proof of Lemma \ref{chirurg} we need to distinguish several
cases. First assume $\eta\notin C(R)$. By Lemma \ref{spezordx2x}
there exists $s\in B'[C]$ with $s(\eta)=0$ and $\ord_\xi(\ol s)=1$.
Let $\tau$ be the automorphism of $R'[C]$ of order two that is
induced by complex conjugation. Then $p_\eta:=s\cdot\tau(s)$ is a sum
of two squares in $B[C]$, and clearly satisfies $p_\eta(\eta)=0$ and
$\ord_\xi(\ol p_\eta)=2$.

When $\eta$ is an interior point of $K_R$, choose $s\in B[C]$ with
$s(\eta)=0$ and $\ord_\xi(\ol s)=1$, see Lemma \ref{spezordx2x}(b).
Then $p_\eta:=s^2$ will do the job.

Now assume that $\eta$ is real but not an interior point of $K_R$.
Then necessarily $\xi$ is a boundary point of $K$. Since $T$ is
saturated, there exists $t\in T$ with $\ord_\xi(t)=1$. As a function
on $C(R)$, $t$ therefore changes sign in $\xi$ and has no other zero
in $U(\xi)$. Since $\eta\in U(\xi)$ is not an interior point of
$K_R$, we have $t(\eta)\le0$.
So $p_\eta:=t-t(\eta)$ lies in $T_B$ and has the desired properties.
With this the lemma is proved, but it remains to add a remark fixing
the isolated point case, see \ref{how2cmpl}. So assume that $\xi$ is
an isolated point of $K$. In this case, since $T$ is saturated, there
is a second element $t'\in T$ with $\ord_\xi(t')=1$ such that $tt'\le
0$ in a neighborhood of $\xi$ (on $K$, or on $K_R$). In the
definition of $p$ in \ref{how2cmpl}, we replace (in the case when
$f(\xi)=0$)
one of the local factors $t$ at $\xi$ by $t'$.
That is, we replace $t^{\ord_\xi(f)}$ by $t^{\ord_\xi(f)-1}t'$. In
this way we have changed the sign of $g(\xi)$ in \ref{how2cmpl}. We
see that we can ensure that we always have $g(\xi)>0$.

This proves Lemma \ref{chirurg}, and this completes the proof of
Theorem \ref{main}.
\qed
\end{lab}

%-------------------------------------------------------------------%

\section{Semidefinite representations in the compact
  case}\label{sect:sdp}%

Now we use moment relaxation to obtain semidefinite representations
from the results of the previous section.

\begin{lab}\label{sdprep}%
Let $K\subset\R^n$ be a compact and convex \sa\ set whose set
$\Ex(K)$ of extremal points has (\sa) dimension at most one. The
closure $K_0=\ol{\Ex(K)}$ is a compact \sa\ set satisfying $\dim(K_0)
\le1$, and $K=\conv(K_0)$ by the Krein-Milman theorem. We assume that
no nonzero linear polynomial vanishes identically on $K$.

Let $C_0\subset\A^n$ be the (reduced) Zariski closure of $K_0$ in
affine $n$-space, so $C_0$ is an $\R$-variety (usually reducible) of
dimension~$\le1$. Let $C'_0\to C_0$ be the normalization of $C_0$. On
$\R$-points, the map $C'_0(\R)\to C_0(\R)$ may fail to be surjective.
Namely, when $\xi$ is an isolated point of $C_0(\R)$ that is singular
as a point on $C_0$, then $\xi$ is not in the image of $C'_0(\R)$.
Let $\xi_1,\dots,\xi_k$ be the isolated points of $C_0(\R)$ that lie
in $K_0$ and are singular points of $C_0$, and write $P_i=\Spec(\R)$
for $i=1,\dots,k$. Finally let
$$C_1\>=\>C'_0\>\amalg\>P_1\>\amalg\>\cdots\>\amalg\>P_k$$
(disjoint sum), and let $\phi\colon C_1\to C_0\subset\A^n$ be the
canonical morphism sending $P_i$ to $\xi_i$ ($i=1,\dots,k$). Let
$K_1$ be the preimage of $K_0$ in $C_1(\R)$. Then $K_1$ is a compact
\sa\ subset of $C_1(\R)$, and we have $\phi(K_1)=K_0$.

If $C_0$ has the irreducible components $X_1,\dots,X_l$, if we write
$A_i=\R[X_i]$ and denote by $A'_i$ the integral closure of $A_i$ in
its quotient field, the coordinate ring of $C_1$ is therefore
$$\R[C_1]\>=\>A'_1\times\dots\times A'_l\times\R\times\cdots\times
\R$$
(with $k$ factors $\R$). The saturated preordering $\scrP(K_1)$ of
$K_1$ in $\R[C_1]$ is finitely generated (see Theorem \ref{satpofg})
and stable (Theorem \ref{mainstab}).

The morphism $C_1\to C_0\subset\A^n$ induces a homomorphism $\R[x_1,
\dots,x_n]\to\R[C_1]$ of $\R$-algebras, which is injective when
restricted to $L:=\spn(1,x_1,\dots,x_n)$, by our assumption on $K$.
We consider $L$ as a linear subspace of $\R[C_1]$. Let $\Sigma=\Sigma
\R[C_1]^2$, and choose $1=h_0,\,h_1,\dots,h_r\in\R[C_1]$ with $\scrP
(K_1)=\Sigma h_0+\cdots+\Sigma h_r$. Since $\scrP(K_1)$ is stable,
there exists a tuple $W=(W_0,\dots,W_r)$ of finite-dimensional
subspaces $W_i\subset\R[C_1]$ such that $L\cap\scrP(K_1)$ is
contained in
$$M_W\>=\>\Sigma_{W_0}+\Sigma_{W_1}h_1+\cdots+\Sigma_{W_r}h_r$$
(c.f.\ \ref{relaxrappl}). By Theorem \ref{relaxsumm} this implies
that we have found a semidefinite representation for $\ol{\conv\>
\phi(K_1)}=K$. We have thus proved:
\end{lab}

\begin{thm}\label{sdp}%
Every compact convex \sa\ set $K\subset\R^n$ with $\dim\Ex(K)\le1$
is a projected spectrahedron. A~semidefinite representation of $K$
can be obtained from a suitable moment relaxation.
\qed
\end{thm}

\begin{example}\label{isolpktbsp}%
To illustrate the construction in \ref{sdprep}, let us consider the
(rational) affine curve $C_0$ with equation
$$y^2+x^2(x-1)(x-2)\>=\>0.$$
The set $C_0(\R)$ is compact and has the origin as an isolated point.
To construct a semidefinite representation for the convex hull $K$ of
$C_0(\R)$, we work in $A_1=A'_0\times\R$, where $A'_0$ is the
integral closure of $A_0=\R[C_0]$, i.e., $A'_0=\R[x,z]/(z^2+(x-1)(x-2))$
(where $y=xz$). Using the elements $1=(1,1)$, $u=(x,0)$, $v=(z,0)$
and $e=(1,0)$ of $A_1$, we let $L=\spn(1,u,uv)$, $W=\spn(1,e,u,v)$
and $U=WW=\spn(1,e,u,v,u^2,uv)$. The relaxation for $K$ obtained from
this data is exact. Using the basis $1-e,e,u,v$ for $W$, we get $K$
as the set of all $(\xi,\eta)\in\R^2$ for which there exist $a,b,c\in
\R$ with
$$\begin{pmatrix}1-c&0&0&0\\0&c&\xi&a\\0&\xi&b&\eta\\0&a&\eta&
3\xi-b-2c\end{pmatrix}\>\succeq\>0.$$
\end{example}

\begin{rem}
Here are a few remarks on possible extensions of Corollary
\ref{mainstab} and Theorem \ref{sdp} to dimensions larger than one.
Corollary \ref{mainstab} says, for $K$ a compact \sa\ set on a
nonsingular affine curve, that the saturated preordering $\scrP(K)$
is (finitely generated and) stable. Unfortunately, according to the
main result of \cite{sch:stab}, there does not exist any \sa\ set $K$
with $\dim(K)>1$ to which this result would extend.

On the other hand, note that the application of moment relaxation to
semidefinite representations of convex hulls requires much less than
saturatedness and stability of some finitely generated preordering.
All that is needed is partial (relative) versions of these
conditions with respect to the space $L$ of linear polynomials, see
Corollary \ref{relaxcpt}. These may still be impossible to satisfy,
but there is more freedom: Given a \sa\ set $K\subset\R^n$, one could
hope to find a morphism $\phi\colon V\to\A^n$ of affine varieties and
a basic closed set $K'\subset V(\R)$ satisfying $\phi(K')=K$ such
that the exactness conditions for the moment relaxation can be
satisfied for $K'$ and for the linear subspace in $\R[V]$ which is
the image of $L$ under $\phi^*$. In \ref{sdprep} above, we have used
this approach, replacing a singular curve by its normalization. In
dimension greater than one, we do not know how far one can expect to
get with such a strategy.
\end{rem}

%-------------------------------------------------------------------%

\section{Semidefinite representations in the general
  case}\label{sect:noncpt}%

Using the compact case, we now establish semidefinite representations
for the closed convex hull of an arbitrary one-dimensional \sa\ set,
and will deduce the dimension two case of the Helton-Nie conjecture.
I am indebted to Tim Netzer who showed me how to obtain semidefinite
representations for noncompact closed convex sets from such
representations for compact sets.

\begin{thm}\label{sdpclosed}%
Let $K\subset\R^n$ be the closed convex hull of a \sa\ set of
dimension $\le1$. Then $K$ has a semidefinite representation.
\end{thm}

\begin{lab}\label{convprep}%
Before we start the proof, we need to recall a few notions on convex
sets and cones, for which we refer to \cite{r}. Given a nonempty
closed convex set $K\subset\R^n$, the \emph{recession cone} of $K$ is
$$\rc(K)\>=\>\{x\in\R^n\colon\>K+x\subset K\},$$
and is a closed convex cone. Note that $\rc(K)$ can also be described
as the set of all existing limits $\lim_{\nu\to\infty}a_\nu x_\nu$ in
$\R^n$, where $x_\nu$ is a sequence in $K$ and $a_\nu$ is a null
sequence of positive real numbers.
The homogenization $K^h$ of $K$ is the closure of the convex cone
$K'=\{(t,tx)\colon t\ge0$, $x\in K\}$ in $\R^{n+1}$, and
$$K^h\>=\>K'\cup\{(0,y)\colon y\in\rc(K)\}.$$
The original set $K$ can clearly be recovered from its homogenization
as $K=\{x\in\R^n\colon(1,x)\in K^h\}$. The extremal rays of $K^h$ are
the rays spanned by points $(1,x)$ with $x\in\Ex(K)$, together with
the rays spanned by points $(0,y)$ where $\R_\plus y$ is an extremal
ray of $\rc(K)$.
\end{lab}

\begin{lab}\label{asydir}%
Let $S\subset\R^n$ be a \sa\ set. A ray $\R_\plus u$ (with $0\ne u\in
\R^n$) will be called an \emph{asymptotic direction of $S$ at
infinity} if there exist continuous \sa\ paths $a(t)$ in $\R_\plus$
and $x(t)$ in $S$ (with $0<t\le1$) such that $a(t)\to0$ and $a(t)x(t)
\to u$ for $t\to0$.
\end{lab}

\begin{prop}\label{rcconvhull}%
Let $S\subset\R^n$ be a nonempty closed \sa\ set, and let $K=
\ol{\conv(S)}$ be its closed convex hull.
\begin{itemize}
\item[(a)]
Each extremal point of $K$ is contained in $S$.
\item[(b)]
Each extremal ray of $\rc(K)$ is an asymptotic direction of $S$ at
infinity.
\end{itemize}
\end{prop}

\begin{proof}
For the proof of both parts we can assume $\rc(K)\cap(-\rc(K))=
\{0\}$.
To prove (a) we are going to show that, whenever $\xi\in K\setminus
\conv(S)$, there exists $0\ne u\in\rc(K)$ with $\xi-u\in\conv(S)$
(and hence $\xi$ is not an extremal point of $K$).
Given $\xi\in K$, there exist continuous \sa\ paths $a_i(t)$ in
$[0,1]$ and $x_i(t)$ in $S$, for $i=0,\dots,n$ and $0<t\le1$, such
that $\sum_{i=0}^na_i(t)\equiv1$, and such that
\begin{equation}\label{limpath}%
x(t)\>=\>\sum_{i=0}^na_i(t)x_i(t)
\end{equation}
converges to $\xi$ for $t\to0$. Note that the limit $\alpha_i=\lim
_{t\to0}a_i(t)$ exists in $[0,1]$ for every $0\le i\le n$, and that
$\sum_i\alpha_i=1$.

We claim that the curves $a_i(t)x_i(t)$ ($i=0,\dots,n$) are bounded
for $t\to0$. Indeed, assume that $a_i(t)x_i(t)$ is unbounded for at
least one index~$i$. Let $q>0$ be the minimal rational number such
that, for every $0\le i\le n$, the curve $t^{-q}a_i(t)x_i(t)$ is
bounded and therefore the limit $u_i:=\lim_{t\to0}t^{-q}a_i(t)x_i(t)$
exists in $\R^n$. For each index $i$ we have $u_i\in\rc(K)$, and $u_i
\ne0$ for at least one index~$i$.
Dividing \eqref{limpath} by $t^q$ gives $\sum_{i=0}^nu_i=0$,
contradicting $\rc(K)\cap(-\rc(K))=\{0\}$.

Since the curves $a_i(t)x_i(t)$ are all bounded, the limits $u_i:=
\lim_{t\to0}a_i(t)x_i(t)$ exist in $\R^n$. If $x_i(t)$ is unbounded
then $\alpha_i=0$ and $u_i\in\rc(K)$.
If $x_i(t)$ is bounded then $\xi_i=\lim_{t\to0}x_i(t)$ exists in $S$,
and $u_i=\alpha_i\xi_i$. Let $y$ denote the sum of the $u_i$ for
those indices $i$ for which $x_i(t)$ is bounded, and let $u$ be the
sum of the remaining $u_i$. Then $y\in\conv(S)$,
$u\in\rc(K)$ and $\xi=y+u$. This proves our assertion.

The proof of (b) is similar to the proof of (a). After making a
translation we can assume $0\in K$. Let $0\ne u\in\rc(K)$. Similar to
\eqref{limpath} we have
$$\frac1tu-w(t)\>=\>\sum_{i=0}^na_i(t)x_i(t)$$
($0<t\le1$) with \sa\ paths $a_i(t)$ in $[0,1]$ and $x_i(t)$ in $S$,
where $\sum_ia_i(t)\equiv0$ and $w(t)$ is a correction term with
$|w(t)|<1$.
Multiplication with $t$ gives
$$u-tw(t)\>=\>\sum_{i=0}^nta_i(t)x_i(t).$$
For $t\to0$, the summands on the right remain bounded, as shown
above.
Therefore the limit $u_i=\lim_{t\to0}ta_i(t)x_i(t)$ exists in $\R^n$
for $i=0,\dots,n$, and $\R_\plus u_i$ is an asymptotic direction of
$S$ at infinity (see \ref{asydir}) if $u_i\ne0$.
From $u=\sum_iu_i$ we see that if $\R_\plus u$ is an extremal ray of
$\rc(K)$, then $\R_\plus u=\R_\plus u_i$ for some $i$, which proves
(b).
\end{proof}

\begin{lab}\label{pfsdpclosed}%
We now give the proof of Theorem \ref{sdpclosed}. Let $S\subset\R^n$
be a nonempty \sa\ set of dimension at most one, and let $K=
\ol{\conv(S)}$ be its closed convex hull. In order to prove that $K$
has a semidefinite representation we may assume $\rc(K)\cap(-\rc(K))
=\{0\}$.
For the homogenization $K^h\subset\R^{n+1}$ of $K$ (see
\ref{convprep}) this implies $K^h\cap(-K^h)=\{0\}$.
So the dual cone $(K^h)^*$ of $K^h$ in $\R^{n+1}$ is
full-dimensional,
and we can pick an interior point $w$ of $(K^h)^*$. The convex set
$$K_1\>:=\>\{x\in K^h\colon\>\bil xw=1\}$$
is compact,
and $K^h$ is (isomorphic to) the homogenization of $K_1$.

The extremal rays of the convex cone $K^h$ correspond to the extremal
points of $K$ and to the extremal rays of $\rc(K)$, see
\ref{convprep}. By Proposition \ref{rcconvhull}, $\Ex(K)\subset\ol S$
has dimension $\le1$, and $\rc(K)$ has only finitely many extremal
rays.
Considering the set of extremal rays of $K^h$ as a subset of the unit
sphere in $\R^{n+1}$, this set therefore has dimension $\le1$. It
follows that the set $\Ex(K_1)$ of extremal points of $K_1$ has
dimension $\le1$ as well. So we can apply Theorem \ref{sdp} to $K_1$,
and conclude that $K_1$ has a semidefinite representation. By Lemma
\ref{kptaffsdp} below, this implies that the cone $(K_1)^h\cong K^h$
has a semidefinite representation as well. This completes the proof
of Theorem \ref{sdpclosed}, since $K$, being an affine-linear section
of $K^h$, also has a semidefinite representation.
\end{lab}

\begin{lem}\label{kptaffsdp}%
Let $K\subset\R^n$ be a compact convex \sa\ set. If $K$ has a
semidefinite representation, then the homogenization $K^h$ of $K$
has a semidefinite representation as well.
\end{lem}

\begin{proof}
Since $K$ is compact, $K^h=\{(t,tx)\colon t\ge0$, $x\in K\}$ is
simply the cone over $K$. Therefore, a semidefinite representation of
$K^h$ is obtained by homogenizing a representation of $K$. In more
detail, assume that
$$K\>=\>\{x\in\R^n\colon\>\ex y\in\R^k\ M(1,x)+N(y)\succeq0\}$$
where $M(t,x)=tM_0+\sum_{i=1}^nx_iM_i$ and $N(y)=\sum_{j=1}^ky_jN_j$
are linear systems of symmetric matrices. Homogenizing the linear
matrix inequality we obtain a semidefinitely representable convex
cone
$$C\>:=\>\{(t,x)\in\R^{n+1}\colon\>t\ge0,\ \ex y\in\R^k\ M(t,x)+N(y)
\succeq0\}$$
in $\R^{n+1}$. We claim that the obvious inclusion $K^h\subset C$ is
an equality. Indeed, both convex cones are contained in the closed
half-space $t\ge0$, and they intersect the open half-space $t>0$ in
the same (nonempty) set. Since $K^h$ is closed, this implies the
reverse inclusion $C\subset K^h$.
\end{proof}

Now we combine Theorem \ref{sdpclosed} with results of Netzer to
show:

\begin{thm}\label{hndim2}%
\emph{(Helton-Nie conjecture in dimension two)}
Every convex \sa\ subset of $\R^2$ is a projected spectrahedron.
\end{thm}

\begin{proof}
Let $K\subset\R^2$ be a convex \sa\ set. To prove that $K$ has a
semidefinite representation, we first consider the case when $K$ is
closed. If $K$ contains a line, the assertion is obvious by reduction
to a (closed) convex subset of $\R$.
So we assume $\rc(K)\cap(-\rc(K))=\{0\}$. Then $K=\conv\Ex(K)+\rc(K)=
\ol{\conv\Ex(K)}+\rc(K)$ (\cite{r} Theorem 18.5),
and the set $\Ex(K)$ is \sa\ of dimension $\le1$. By Theorem
\ref{sdpclosed}, $\ol{\conv\Ex(K)}$ is a projected spectrahedron.
Since $\rc(K)$ is clearly a projected spectrahedron,
we see that $K$ is one as well.

Now let $K\subset\R^2$ be an arbitrary convex \sa\ set. We can assume
that $K$ has nonempty interior. Let $M$ be the set of points in the
boundary $\partial K=\partial\ol K$ that do not lie in $K$.
Then $M$ is a \sa\ set with $\dim(M)\le1$, and we can decompose $M$
set-theoretically as follows. Let $M_0$ be the relative interior of
$M\cap\Ex(\ol K)$ inside $\partial\ol K$, and let $\scrF$ be the
(finite) set of one-dimensional faces of $\ol K$.
For each $F\in\scrF$, let $M_F=F\cap M$. Moreover, let $H_F$ be the
open halfplane with $H_F\cap K\ne\emptyset$ whose boundary line
contains $F$, and let $K_F=H_F\cup(F\cap K)=H_F\cup(\ol H_F\cap K)$.
Then $M$ is the union of $M_0$ with finitely many extremal points of
$\ol K$ and with $\bigcup_{F\in\scrF}M_F$.
Accordingly, $K$ is the intersection of $K_0:=\ol K\setminus M_0$
with finitely many sets $K_\xi:=\ol K\setminus\{\xi\}$ (where $\xi\in
\Ex(\ol K)$) and with the sets $K_F$ ($F\in\scrF$).

Since a finite intersection of projected spectrahedra is again a
projected spectrahedron, it suffices to show that each of $K_0$,
$K_\xi$ and $K_F$ as above is a projected spectrahedron. Each of the
sets $K_F$ is a union of an open halfplane $H$ with a convex subset
of the line $\partial H$. Using the result of \cite{ns}, such $K_F$
is a projected spectrahedron. (Due to the elementary nature of this
situation, one can easily find an explicit semidefinite
representation directly.)
The sets $K_\xi$ ($\xi\in\Ex(\ol K)$) are projected spectrahedra by
\cite{nt} Proposition 3.1.
For $K_0$ we use Netzer's construction from \cite{nt}. Let $N=
\partial\ol K\setminus M_0$, a closed subset of $\partial\ol K$ with
$K_0=\interior(K)\cup N$, and let $T=\ol{\conv(N)}$. Then $T$ is a
closed convex subset of $\ol K$, and is a projected spectrahedron by
Theorem \ref{sdpclosed}. By construction, and by Proposition
\ref{rcconvhull}, $T\cap\partial\ol K=N=\partial\ol K\setminus M_0$.
Using the notation introduced in \cite{nt}, let $(T\lal\ol K)$ denote
the union of the relative interiors of all the faces of $\ol K$ that
meet $T$. We see that $(T\lal\ol K)=\interior(K)\cup N=K_0$. By
\cite{nt} Theorem 3.8, $(T\lal\ol K)$ is a projected spectrahedron,
which proves our theorem.
\end{proof}

%-------------------------------------------------------------------%

\section{Stability in the virtually compact case}%
\label{sect:virtcpt}%

\begin{lab}\label{virtcptrappl}%
Let $C$ be an irreducible affine curve over $\R$, and let $K\subset
C(\R)$ be a closed \sa\ subset. Adopting the terminology of
\cite{sch:mz}, \cite{sch:guide}, we say that $K$ is \emph{virtually
compact} if there exists an irreducible affine curve $C_1$ containing
$C$ as a Zariski open subset, in such a way that the points in $C_1
\setminus C$ are nonsingular on $C_1$ and the closure $K_1$ of $K$ in
$C_1(\R)$ is compact. Equivalently, $K$ is virtually compact if and
only if there exists a nonconstant regular function $f\in\R[C]$ that
is bounded on $K$.

When the affine curve $C$ is not necessarily irreducible, a closed
\sa\ set $K\subset C(\R)$ is called virtually compact if $K\cap
C'(\R)$ is virtually compact on $C'$, for every irreducible component
$C'$ of~$C$. A closed \sa\ set $K\subset\R^n$ of dimension $\le1$ is
called virtually compact if it has this property with respect to its
Zariski closure $C$.

We show that the analogues of the stability results from Section
\ref{sect:main} remain true for virtually compact sets $K$:
\end{lab}

\begin{thm}\label{vcptmain}%
Let $C$ be an irreducible nonsingular affine curve over $\R$, and let
$K\subset C(\R)$ be a closed \sa\ subset that is virtually compact.
Then the saturated preordering $\scrP(K)$ in $\R[C]$ is finitely
generated and stable.
\end{thm}

\begin{proof}
That $\scrP(K)$ is finitely generated was already proved (in greater
generality) in \cite{sch:mz} Theorem 5.21. We are going to reprove
this fact here using a different reasoning, because we'll need the
same argument to prove stability. Since the theorem has already been
proved when $K$ is compact, we can assume that $K$ is not compact. In
particular, the set $K$ is infinite.

Let $C_1$ and $K_1$ be as in \ref{virtcptrappl}. Then $\R[C_1]$ is a
subring of $\R[C]$. Let $T=\scrP_C(K)$ (the saturated preordering of
$K$ in $\R[C]$), and let $T_1=\scrP_{C_1}(K_1)$ (the saturated
preordering of $K_1$ in $\R[C_1]$). Since $K_1$ is compact, the
preordering $T_1$ in $\R[C_1]$ is finitely generated, according to
Theorem \ref{satpofg}.
So there are nonzero elements $1=h_0,\,h_1,\dots,h_r\in\R[C_1]$ such
that $T_1$ is the quadratic module in $\R[C_1]$ generated by $h_1,
\dots,h_r$. (We can even do with $r\le2$, see \ref{satpofgpf}.) We'll
prove that $T=\scrP_C(K)$ is generated by $h_1,\dots,h_r$ in $\R[C]$.

Let $\wt C$ be the nonsingular projective curve over $\R$ that
contains $C_1$ as an open dense subscheme. We consider Weil divisors
on $\wt C$, and regard them as conjugation-invariant Weil divisors on
the complexified curve $\wt C_\C$. Since $\wt C(\R)\ne\emptyset$,
we have $\Pic(\wt C)=\Pic(\wt C_\C)^{\Gal(\C/\R)}$.
Let $J$ be the Jacobian variety of $\wt C$, an abelian variety over
$\R$.

Let $C_1(\C)\setminus C(\C)=\{Q_1,\dots,Q_s\}$, and let $0\ne f\in
\R[C]$ with $f|_K\ge0$. For $i=1,\dots,s$ let $m_i\ge0$ be an integer
satisfying $2m_i+\ord_{Q_i}(f)\ge0$, and consider the divisor
$$D\>=\>\sum_{i=1}^sm_iQ_i$$
on $\wt C$. Choose a point $Q\in\wt C(\C)\setminus C_1(\C)$, and let
$E=Q+\ol Q$ (again a divisor on $\wt C$, the case $Q=\ol Q$ is
allowed, bar denoting complex conjugation). There exist integers
$l$, $n\ge1$ such that the divisor $nE-lD$ has degree zero, and such
that the divisor class $[nE-lD]\in J(\R)$ lies in the identity
connected component $J(\R)_0$ of the compact real Lie group $J(\R)$.
Fix an arbitrary $\R$-point $P_0$ in the interior $\interior(K)$ of
$K$ relative to $C(\R)$. By the argument in \cite{sch:tams}, 2.11 and
2.12, there is an integer $k\ge1$ such that, for every $\alpha\in
J(\R)_0$, there exist $2k$ points $P_1,\dots,P_{2k}\in\interior(K)$
with
$$\alpha\>=\>\sum_{j=1}^{2k}[P_j-P_0].$$
Applying this to the divisor class $\alpha:=[nE-lD-k(2P_0-E)]$ (which
lies in $J(\R)_0$, c.f.\ \cite{sch:tams} Lemma 2.6), we conclude that
there exist $P_1,\dots,P_{2k}\in\interior(K)$ such that
$$lD+\sum_{j=1}^{2k}P_j\>\sim\>(n+k)E$$
on $\wt C$. Since $\supp(E)$ is disjoint to $C_1$, there exists $0\ne
h\in\R[C_1]$ such that the divisor of $h$ on $C_1$ is $lD+\sum_{j=1}
^{2k}P_j$. Since $\ord_{Q_i}(h^2f)\ge2lm_i+\ord_{Q_i}(f)\ge0$, we see
that $h^2f$ lies in $\R[C_1]$. Moreover, every zero of $h$ on $C$ is
real and is an interior point of $K$.
In addition, we can ensure that $h$ has no common zero with any of
$h_0,\dots,h_r$.

Since $f\ge0$ on $K$, and since $K$ is dense in $K_1$, it follows
that $h^2f\ge0$ on $K_1$. So $h^2f\in T_1$, which means that there is
an identity
$$h^2f\>=\>\sum_{i=0}^r\sum_jp_{ij}^2h_i$$
with suitable $p_{ij}\in\R[C_1]$. Since any zero of $h$ on $C$ is real
and is an interior point of $K$, it follows that each summand
$p_{ij}^2h_i$ of the right hand sum is divisible (inside $\R[C]$) by
$h^2$, see \cite{sch:tams} Lemma 0.1. By the choice of $h$, none of
the $h_i$ vanishes in any of the zeros of $h$. Hence we even have
$h\mid p_{ij}$ (inside $\R[C]$), for all indices $i,\,j$. Dividing we
conclude that $f$ lies in the quadratic module generated by $h_0,
\dots,h_r$ in $\R[C]$.

We have thus proved that $T=\scrP_C(K)$ is finitely generated in
$\R[C]$. To prove that $T$ is stable is equivalent to proving the
following assertion (c.f.\ \cite{sch:stab} Corollary 3.8): Let $R$
be any real closed extension field of $\R$. Then the preordering
$T_R$ generated by $T$ in $R[C]$ is saturated.

To prove this, let $0\ne f\in R[C]$ be nonnegative on $K_R$, where
$K_R$ denotes the extension of the \sa\ set $K\subset C(\R)$ to a
\sa\ subset of $C(R)$. Arguing literally as in the first part of the
proof, we find
$h\in\R[C_1]$ such that $h^2f\in R[C_1]$, and such that any zero of
$h$ on $C$ is real and is an interior point of $K$ in which
$h_0\cdots h_r$ does not vanish. Completing the argument exactly as
before, we see that $f$ lies in the quadratic module of $R[C]$
generated by $h_0,\dots,h_r$. In other words, $f\in T_R$, as desired.
The theorem is proved.
\end{proof}

\begin{rems}
\hfil\smallskip

1.\
The proof has shown that $T=\scrP_C(K)$ is the preordering generated
by the preordering $T_1=\scrP_{C_1}(K_1)$ of $\R[C_1]$ in the larger
ring $\R[C]$. The same fact is readily deduced from \cite{sch:mz}
Theorem 5.5.
\smallskip

2.\
Without using Theorem \ref{sdpclosed}, we can conclude from Theorem
\ref{vcptmain}: For any closed \sa\ set $K\subset\R^n$ of dimension
$\le1$ which is virtually compact, the closed convex hull
$\ol{\conv(K)}$ is the closure of a projected spectrahedron. Indeed,
by arguing exactly as in \ref{sdprep} one finds a projected
spectrahedron sandwiched between $\conv(K)$ and $\ol{\conv(K)}$. Note
however that this conclusion is weaker than Theorem \ref{sdpclosed}
in several respects.
\end{rems}

%-------------------------------------------------------------------%

\begin{thebibliography}{SM2}

\bibitem{bcr}
J.~Bochnak, M.~Coste, M.-F.~Roy: \emph{Real Algebraic Geometry}.
Erg.\ Math.\ Grenzgeb.\ (3) {\bf36}, Springer, Berlin, 1998.

\bibitem{bpt}
G.~Blekherman, P.~Parrilo, R.~Thomas:
\emph{Semidefinite Optimization and Convex Algebraic Geometry}.
Forthcoming book, to appear in the MOS-SIAM Series on Optimization,
see \texttt{www.math.washington.edu/\~{}thomas/frg/book.html}.

\bibitem{gn}
J. Gouveia, T. Netzer:
Positive polynomials and projections of spectrahedra.
SIAM J. Optim.\ \textbf{21}, 960--976 (2012).

\bibitem{gpt}
J. Gouveia, P. Parrilo, R. Thomas:
Theta bodies for polynomial ideals.
SIAM J. Optim.\ \textbf{20}, 2097--2118 (2010).

\bibitem{hn1}
W. Helton, J. Nie:
Sufficient and necessary conditions for semidefinite representability
of convex hulls and sets.
SIAM J. Optim.\ \textbf{20}, 759--791 (2009).

\bibitem{hn2}
W. Helton, J. Nie:
Semidefinite representation of convex sets.
Math.\ Program.\ \textbf{122} (Ser.~A), 21--64 (2010).

\bibitem{he}
D. Henrion:
Semidefinite representation of convex hulls of rational varieties.
Acta Appl.\ Math.\ \textbf{115}, 319-327 (2011).

\bibitem{la}
J.\,B. Lasserre:
Convex sets with semidefinite representation.
Math.\ Program.\ \textbf{120} (Ser.~A), 457--477 (2009).

\bibitem{la:book}
J.\,B. Lasserre:
\emph{Moments, Positive Polynomials and Their Applications}.
Imperial College Press, London, 2010.

\bibitem{ma}
M. Marshall:
\emph{Positive Polynomials and Sums of Squares}.
Mathematical Surveys and Monographs \textbf{146}, AMS, Providence,
RI, 2008.

\bibitem{nm}
A. Nemirovski:
Advances in convex optimization: Conic programming.
Int.\ Cong.\ Math.\ vol.~I, European Math.\ Soc., Z\"urich, 2007,
pp.\ 413-�444.

\bibitem{nt}
T. Netzer:
On semidefinite representations of non-closed sets.
Linear Algebra Appl.\ \textbf{432}, 3072--3078 (2010).

\bibitem{nps}
T. Netzer, D. Plaumann, M. Schweighofer:
Exposed faces of semidefinite representable sets.
SIAM J. Optim.\ \textbf{20}, 1944--1955 (2010).

\bibitem{ns}
T. Netzer, R. Sinn:
A note on the convex hull of finitely many projections of
spectrahedra.
Preprint, 2009,
\texttt{arxiv:0908.3386}.

\bibitem{ni}
J. Nie:
First order conditions for semidefinite representations of convex
sets defined by rational or singular polynomials.
Math.\ Program.\ (to appear),
\texttt{arxiv:0806.4721}.

\bibitem{nipast}
J. Nie, P.~Parrilo, B.~Sturmfels:
Semidefinite representation of the $k$-ellipse.
In: \emph{Algorithms in Algebraic Geometry}, A.~Dickenstein,
F-O.~Schreyer, A.~Sommese (eds.), Springer, New York, 2008,
pp.\ 117--132.

\bibitem{pa}
P. Parrilo:
Exact semidefinite representation for genus zero curves.
Talk at the workshop \emph{Positive polynomials and optimization},
Banff, October 8--12, 2006.

\bibitem{ps}
V.~Powers, C.~Scheiderer:
The moment problem for non-compact semialgebraic sets.
Adv.\ Geometry \textbf{1}, 71--88 (2001).

\bibitem{pd}
A.~Prestel, Ch.~N.~Delzell:
\emph{Positive Polynomials}.
Monographs in Mathematics, Springer, Berlin, 2001.

\bibitem{r}
R.\,T. Rockafellar:
\emph{Convex Analysis}.
Princeton University Press, Princeton NJ, 1970.

\bibitem{rs1}
K.~Ranestad, B.~Sturmfels:
On the convex hull of a space curve.
Adv.\ Geometry \textbf{12}, 157--178 (2012).

\bibitem{rs2}
K.~Ranestad, B.~Sturmfels:
The convex hull of a variety.
In: \emph{Notions of Positivity and the Geometry of Polynomials},
P.~Br\"and\'en, M.~Passare, M.~Putinar (eds.), Springer, Basel, 2011,
pp.\ 331--344.

\bibitem{sch:tams}
C.~Scheiderer:
Sums of squares of regular functions on real algebraic varieties.
Trans.\ Am.\ Math.\ Soc.\ \textbf{352}, 1039--1069 (1999).

\bibitem{sch:local}
C.~Scheiderer:
On sums of squares in local rings.
J.~reine angew.\ Math.\ \textbf{540}, 205--227 (2001).

\bibitem{sch:mz}
C.~Scheiderer:
Sums of squares on real algebraic curves.
Math.~Z. \textbf{245}, 725--760 (2003).

\bibitem{sch:surf}
C.~Scheiderer:
Sums of squares on real algebraic surfaces.
Manuscr.~math.\ \textbf{119}, 395--410 (2006).

\bibitem{sch:stab}
C.~Scheiderer:
Non-existence of degree bounds for weighted sums of squares
representations.
J.~Complexity \textbf{21} 823--844 (2005).

\bibitem{sch:guide}
C. Scheiderer:
Positivity and sums of squares: A guide to recent results.
In: \emph{Emerging Applications of Algebraic Geometry}, IMA Vol.\
Math.\ Appl.\ \textbf{149}, Springer, New York, 2009, pp.\ 271--324.

\bibitem{sch:lgp1}
C.~Scheiderer:
Weighted sums of squares in local rings and their completions,~I.
Math.~Z. \textbf{266}, 1--19 (2010).

\bibitem{sch:conv}
C.~Scheiderer:
Convex hulls of curves of genus one.
Adv.\ Math.\ \textbf{228}, 2606--2622 (2011).

\bibitem{sm}
K.~Schm\"udgen: The $K$-moment problem for compact semi-algebraic
sets. Math.\ Ann.\ \textbf{289}, 203--206 (1991).

\end{thebibliography}
\end{document}

1208.3865.extracted_bib
\begin{thebibliography}{SM2}

\bibitem{bcr}
J.~Bochnak, M.~Coste, M.-F.~Roy: \emph{Real Algebraic Geometry}.
Erg.\ Math.\ Grenzgeb.\ (3) {\bf36}, Springer, Berlin, 1998.

\bibitem{bpt}
G.~Blekherman, P.~Parrilo, R.~Thomas:
\emph{Semidefinite Optimization and Convex Algebraic Geometry}.
Forthcoming book, to appear in the MOS-SIAM Series on Optimization,
see \texttt{www.math.washington.edu/\~{}thomas/frg/book.html}.

\bibitem{gn}
J. Gouveia, T. Netzer:
Positive polynomials and projections of spectrahedra.
SIAM J. Optim.\ \textbf{21}, 960--976 (2012).

\bibitem{gpt}
J. Gouveia, P. Parrilo, R. Thomas:
Theta bodies for polynomial ideals.
SIAM J. Optim.\ \textbf{20}, 2097--2118 (2010).

\bibitem{hn1}
W. Helton, J. Nie:
Sufficient and necessary conditions for semidefinite representability
of convex hulls and sets.
SIAM J. Optim.\ \textbf{20}, 759--791 (2009).

\bibitem{hn2}
W. Helton, J. Nie:
Semidefinite representation of convex sets.
Math.\ Program.\ \textbf{122} (Ser.~A), 21--64 (2010).

\bibitem{he}
D. Henrion:
Semidefinite representation of convex hulls of rational varieties.
Acta Appl.\ Math.\ \textbf{115}, 319-327 (2011).

\bibitem{la}
J.\,B. Lasserre:
Convex sets with semidefinite representation.
Math.\ Program.\ \textbf{120} (Ser.~A), 457--477 (2009).

\bibitem{la:book}
J.\,B. Lasserre:
\emph{Moments, Positive Polynomials and Their Applications}.
Imperial College Press, London, 2010.

\bibitem{ma}
M. Marshall:
\emph{Positive Polynomials and Sums of Squares}.
Mathematical Surveys and Monographs \textbf{146}, AMS, Providence,
RI, 2008.

\bibitem{nm}
A. Nemirovski:
Advances in convex optimization: Conic programming.
Int.\ Cong.\ Math.\ vol.~I, European Math.\ Soc., Z\"urich, 2007,
pp.\ 413-�444.

\bibitem{nt}
T. Netzer:
On semidefinite representations of non-closed sets.
Linear Algebra Appl.\ \textbf{432}, 3072--3078 (2010).

\bibitem{nps}
T. Netzer, D. Plaumann, M. Schweighofer:
Exposed faces of semidefinite representable sets.
SIAM J. Optim.\ \textbf{20}, 1944--1955 (2010).

\bibitem{ns}
T. Netzer, R. Sinn:
A note on the convex hull of finitely many projections of
spectrahedra.
Preprint, 2009,
\texttt{arxiv:0908.3386}.

\bibitem{ni}
J. Nie:
First order conditions for semidefinite representations of convex
sets defined by rational or singular polynomials.
Math.\ Program.\ (to appear),
\texttt{arxiv:0806.4721}.

\bibitem{nipast}
J. Nie, P.~Parrilo, B.~Sturmfels:
Semidefinite representation of the $k$-ellipse.
In: \emph{Algorithms in Algebraic Geometry}, A.~Dickenstein,
F-O.~Schreyer, A.~Sommese (eds.), Springer, New York, 2008,
pp.\ 117--132.

\bibitem{pa}
P. Parrilo:
Exact semidefinite representation for genus zero curves.
Talk at the workshop \emph{Positive polynomials and optimization},
Banff, October 8--12, 2006.

\bibitem{ps}
V.~Powers, C.~Scheiderer:
The moment problem for non-compact semialgebraic sets.
Adv.\ Geometry \textbf{1}, 71--88 (2001).

\bibitem{pd}
A.~Prestel, Ch.~N.~Delzell:
\emph{Positive Polynomials}.
Monographs in Mathematics, Springer, Berlin, 2001.

\bibitem{r}
R.\,T. Rockafellar:
\emph{Convex Analysis}.
Princeton University Press, Princeton NJ, 1970.

\bibitem{rs1}
K.~Ranestad, B.~Sturmfels:
On the convex hull of a space curve.
Adv.\ Geometry \textbf{12}, 157--178 (2012).

\bibitem{rs2}
K.~Ranestad, B.~Sturmfels:
The convex hull of a variety.
In: \emph{Notions of Positivity and the Geometry of Polynomials},
P.~Br\"and\'en, M.~Passare, M.~Putinar (eds.), Springer, Basel, 2011,
pp.\ 331--344.

\bibitem{sch:tams}
C.~Scheiderer:
Sums of squares of regular functions on real algebraic varieties.
Trans.\ Am.\ Math.\ Soc.\ \textbf{352}, 1039--1069 (1999).

\bibitem{sch:local}
C.~Scheiderer:
On sums of squares in local rings.
J.~reine angew.\ Math.\ \textbf{540}, 205--227 (2001).

\bibitem{sch:mz}
C.~Scheiderer:
Sums of squares on real algebraic curves.
Math.~Z. \textbf{245}, 725--760 (2003).

\bibitem{sch:surf}
C.~Scheiderer:
Sums of squares on real algebraic surfaces.
Manuscr.~math.\ \textbf{119}, 395--410 (2006).

\bibitem{sch:stab}
C.~Scheiderer:
Non-existence of degree bounds for weighted sums of squares
representations.
J.~Complexity \textbf{21} 823--844 (2005).

\bibitem{sch:guide}
C. Scheiderer:
Positivity and sums of squares: A guide to recent results.
In: \emph{Emerging Applications of Algebraic Geometry}, IMA Vol.\
Math.\ Appl.\ \textbf{149}, Springer, New York, 2009, pp.\ 271--324.

\bibitem{sch:lgp1}
C.~Scheiderer:
Weighted sums of squares in local rings and their completions,~I.
Math.~Z. \textbf{266}, 1--19 (2010).

\bibitem{sch:conv}
C.~Scheiderer:
Convex hulls of curves of genus one.
Adv.\ Math.\ \textbf{228}, 2606--2622 (2011).

\bibitem{sm}
K.~Schm\"udgen: The $K$-moment problem for compact semi-algebraic
sets. Math.\ Ann.\ \textbf{289}, 203--206 (1991).

\end{thebibliography}