### A conic and an $M$-quintic with a point at infinity.

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Let $C$ be a smooth real quartic curve in ${\mathbb{P}}^{2}$. Suppose that $C$ has at least $3$ real branches ${B}_{1},{B}_{2},{B}_{3}$. Let $B={B}_{1}\times {B}_{2}\times {B}_{3}$ and let $O\in B$. Let ${\tau}_{O}$ be the map from $B$ into the neutral component Jac$\left(C\right){\left(\mathbb{R}\right)}^{0}$ of the set of real points of the jacobian of $C$, defined by letting ${\tau}_{O}\left(P\right)$ be the divisor class of the divisor $\sum {P}_{i}-{O}_{i}$. Then, ${\tau}_{O}$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac$\left(C\right){\left(\mathbb{R}\right)}^{0}$. It generalizes the classical geometric description of the group law on the neutral component of the set of real points of...

Classical Lüroth theorem states that every subfield K of K(t), where t is a transcendental element over K, such that K strictly contains K, must be K = K(h(t)), for some non constant element h(t) in K(t). Therefore, K is K-isomorphic to K(t). This result can be proved with elementary algebraic techniques, and therefore it is usually included in basic courses on field theory or algebraic curves. In this paper we study the validity of this result under weaker assumptions: namely, if K is a subfield...

We show that for a polynomial mapping $F=(f\u2081,...,f\u2098):{\u2102}^{n}\to {\u2102}^{m}$ the Łojasiewicz exponent ${}_{\infty}\left(F\right)$ of F is attained on the set $z\in {\u2102}^{n}:f\u2081\left(z\right)\xb7...\xb7f\u2098\left(z\right)=0$.

Investigated are continuous rational maps of nonsingular real algebraic varieties into spheres. In some cases, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular, each continuous map between unit spheres can be approximated by continuous rational maps.

We show that in the class of compact, piecewise ${C}^{1}$ curves K in ${\mathbb{R}}^{n}$, the semialgebraic curves are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for the derivatives of (the traces of) polynomials on K.

We show that in the class of compact sets K in ${\mathbb{R}}^{n}$ with an analytic parametrization of order m, the sets with Zariski dimension m are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for tangential derivatives of (the traces of) polynomials on K.