F. Cukierman asked whether or not for every smooth real plane curve $X\subset \mathbb{P}^{2}$ of even degree $d\geqslant 2$ there exists a real line $L\subset \mathbb{P}^{2}$ such $X\cap L$ has no real points. We show that the answer is yes if $d=2$ or 4 and no if $n\geqslant 6$.

We define Schwartz functions, tempered functions, and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds, also hold in the case of affine real algebraic varieties, and give partial results for the non-affine case.

We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in $\mathbb{P}^{3}$ whose function field has level 2 is dense in the set of those that have no real points.

We present a new approach to noncommutative real algebraic geometry based on the representation theory of ${{C}^{*}}$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative ${{C}^{*}}$-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.