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.

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}.

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 $X$.

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.

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.

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.