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1. CMB 2009 (vol 52 pp. 224)
Equations and Complexity for the Dubois--Efroymson Dimension Theorem Let $\R$ be a real closed field, let $X \subset \R^n$ be an
irreducible real algebraic set and let $Z$ be an algebraic subset of
$X$ of codimension $\geq 2$. Dubois and Efroymson proved the existence
of an irreducible algebraic subset of $X$ of codimension $1$
containing~$Z$. We improve this dimension theorem as follows. Indicate
by $\mu$ the minimum integer such that the ideal of polynomials in
$\R[x_1,\ldots,x_n]$ vanishing on $Z$ can be generated by polynomials
of degree $\leq \mu$. We prove the following two results:
\begin{inparaenum}[\rm(1)]
\item There
exists a polynomial $P \in \R[x_1,\ldots,x_n]$ of degree~$\leq \mu+1$
such that $X \cap P^{-1}(0)$ is an irreducible algebraic subset of $X$
of codimension $1$ containing~$Z$.
\item Let $F$ be a polynomial in
$\R[x_1,\ldots,x_n]$ of degree~$d$ vanishing on $Z$. Suppose there
exists a nonsingular point $x$ of $X$ such that $F(x)=0$ and the
differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then
there exists a polynomial $G \in \R[x_1,\ldots,x_n]$ of degree $\leq
\max\{d,\mu+1\}$ such that, for each $t \in (-1,1) \setminus \{0\}$,
the set $\{x \in X \mid F(x)+tG(x)=0\}$ is an irreducible algebraic
subset of $X$ of codimension $1$ containing~$Z$.
\end{inparaenum} Result (1) and a
slightly different version of result~(2) are valid over any
algebraically closed field also.
Keywords:Irreducible algebraic subvarieties, complexity of algebraic varieties, Bertini's theorems Categories:14P05, 14P20 |
2. CMB 2005 (vol 48 pp. 90)
Products of Conjugacy Classes in $SU(2)$ We obtain a complete description of the conjugacy classes
$C_1,\dots,C_n$ in $SU(2)$ with the property that $C_1\cdots
C_n=SU(2)$. The basic instrument is a characterization of the
conjugacy classes $C_1,\dots,C_{n+1}$ in $SU(2)$ with $C_1\cdots
C_{n+1}\ni I$, which generalizes a result of \cite{Je-We}.
Categories:14D20, 14P05 |
3. CMB 2001 (vol 44 pp. 257)
Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces Let $X$ be a reduced nonsingular quasiprojective scheme over ${\mathbb
R}$ such that the set of real rational points $X({\mathbb R})$ is dense
in $X$ and compact. Then $X({\mathbb R})$ is a real algebraic variety.
Denote by $H_k^{\alg}(X({\mathbb R}), {\mathbb Z}/2)$ the group of
homology classes represented by Zariski closed $k$-dimensional
subvarieties of $X({\mathbb R})$. In this note we show that $H_1^{\alg}
(X({\mathbb R}), {\mathbb Z}/2)$ is a proper subgroup of
$H_1(X({\mathbb R}), {\mathbb Z}/2)$ for a nonorientable hyperelliptic
surface $X$. We also determine all possible groups $H_1^{\alg}(X({\mathbb R}),
{\mathbb Z}/2)$ for a real ruled surface $X$ in connection with the previously
known description of all possible topological configurations of $X$.
Categories:14P05, 14P25 |
4. CMB 1999 (vol 42 pp. 445)
Smooth Maps and Real Algebraic Morphisms Let $X$ be a compact nonsingular real algebraic variety and let $Y$
be either the blowup of $\mathbb{P}^n(\mathbb{R})$ along a linear
subspace or a nonsingular hypersurface of $\mathbb{P}^m(\mathbb{R})
\times \mathbb{P}^n(\mathbb{R})$ of bidegree $(1,1)$. It is proved
that a $\mathcal{C}^\infty$ map $f \colon X \rightarrow Y$ can be
approximated by regular maps if and only if $f^* \bigl( H^1(Y,
\mathbb{Z}/2) \bigr) \subseteq H^1_{\alg} (X,\mathbb{Z}/2)$, where
$H^1_{\alg} (X,\mathbb{Z}/2)$ is the subgroup of $H^1 (X,
\mathbb{Z}/2)$ generated by the cohomology classes of algebraic
hypersurfaces in $X$. This follows from another result on maps
into generalized flag varieties.
Categories:14P05, 14P25 |
5. CMB 1999 (vol 42 pp. 307)
On the Moduli Space of a Spherical Polygonal Linkage We give a ``wall-crossing'' formula for computing the topology of
the moduli space of a closed $n$-gon linkage on $\mathbb{S}^2$.
We do this by determining the Morse theory of the function
$\rho_n$ on the moduli space of $n$-gon linkages which is given by
the length of the last side---the length of the last side is
allowed to vary, the first $(n - 1)$ side-lengths are fixed. We
obtain a Morse function on the $(n - 2)$-torus with level sets
moduli spaces of $n$-gon linkages. The critical points of $\rho_n$
are the linkages which are contained in a great circle. We give a
formula for the signature of the Hessian of $\rho_n$ at such a
linkage in terms of the number of back-tracks and the winding
number. We use our formula to determine the moduli spaces of all
regular pentagonal spherical linkages.
Categories:14D20, 14P05 |
6. CMB 1997 (vol 40 pp. 456)
Approximation of smooth maps by real algebraic morphisms Let $\Bbb G_{p,q}(\Bbb F)$ be the Grassmann space of all
$q$-dimensional $\Bbb F$-vector subspaces of $\Bbb F^{p}$, where $\Bbb F$
stands for $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). Here
$\Bbb G_{p,q}(\Bbb F)$ is regarded as a real algebraic variety. The paper
investigates which ${\cal C}^\infty$ maps from a nonsingular real algebraic
variety $X$ into $\Bbb G_{p,q}(\Bbb F)$ can be approximated, in the
${\cal C}^\infty$ compact-open topology, by real algebraic morphisms.
Categories:14P05, 14P25 |