Let $X$, $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$-regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$, we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$-regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$-regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$-dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$-regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$.

This paper solves the rational noncommutative analogue of Hilbert’s 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of Hermitian matrices in its domain, then it is a sum of Hermitian squares of noncommutative rational functions. This result is a generalisation and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without Hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $L\succeq 0$ if and only if it belongs to the rational quadratic module generated by L. The essential intermediate step toward this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils.

The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with a certain symmetry.

A class of rational functions and sufficient conditions on it such that the Julia set is a Jordan arc, respectively no Jordan arc, are introduced. Further, it is shown that any rational function which has a Jordan arc as Julia set belongs up to Möbius conjugation to this class.

This paper considers the rational system ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})\,\,:=\,\left\{ \frac{P(x)}{\Pi _{k=1}^{n}(x-{{a}_{k}})},\,P\,\in \,{{P}_{n}} \right\}$ with nonreal elements in $\left\{ {{a}_{k}} \right\}_{k=1}^{n}\,\subset \,\mathbb{C}\,\backslash \,[-1,\,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$. The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$ are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results