In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let $A_1,...,A_m\subseteq {ℝ}^n$ be subsets with finite Lebesgue measure. Then, for any sequence $f_0,...,f_m$ of $ℝ$-linearly independent polynomials in the polynomial ring $ℝ[X_1,...,X_n]$ there are real numbers ${\lambda }_0,...,{\lambda }_m$, not all zero, such that the real affine variety $\{x\in {ℝ}^n;{\lambda }_0{f}_0\left(x\right)+\cdots +{\lambda }_m{f}_m\left(x\right)=0\}$ simultaneously bisects each of subsets $A_k$, $k=1,...,m$. Then some its applications are studied.

Let Y be a real algebraic subset of ${}^m$ and $F:Y{\to }^n$ be a polynomial map. We show that there exist real polynomial functions $g_1,...,g_s$ on ${}^n$ such that the Euler characteristic of fibres of $F$ is the sum of signs of $g_i$.

The paper is concerned with an effective formula for the Euler characteristic of the link of a weighted homogeneous mapping $F:ℝⁿ\to {ℝ}^k$ with an isolated singularity. The formula is based on Szafraniec’s method for calculating the Euler characteristic of a real algebraic manifold (as the signature of an appropriate bilinear form). It is shown by examples that in the case of a weighted homogeneous mapping it is possible to make the computer calculations of the Euler characteristics much more effective.

Let V be a real algebraic manifold of positive dimension. The aim of this paper is to show that, for every integer b (arbitrarily large), there exists a trivial Nash family $={V_y}_{y\in R^b}$ of real algebraic manifolds such that V₀ = V, is an algebraic family of real algebraic manifolds over $y\in R^b\setminus 0$ (possibly singular over y = 0) and is perfectly parametrized by $R^b$ in the sense that $V_y$ is birationally nonisomorphic to $V_z$ for every $y,z\in R^b$ with y ≠ z. A similar result continues to hold if V is a singular real algebraic set.

We introduce a notion of generic real algebraic variety and we study the space of morphisms into these varieties. Let $Z$ be a real algebraic variety. We say that $Z$ is generic if there exist a finite family ${\left\{D_i\right\}}_{i=1}^n$ of irreducible real algebraic curves with genus $\ge 2$ and a biregular embedding of $Z$ into the product variety ${\prod }_{i=1}^n{D}_i$. A bijective map $\varphi :{\tilde{Z}}^{\phantom{1}}\to Z$ from a real algebraic variety $\tilde{Z}$ to $Z$ is called weak change of the algebraic structure of $Z$ if it is regular and its inverse is a Nash map. Generic real algebraic varieties...

In this Note, we announce several results concerning basic properties of the spaces of morphisms between real algebraic varieties. Our results show a surprising intrinsic rigidity of Real Algebraic Geometry and illustrate the great distance which, in some sense, exists between this geometry and Real Nash one. Let us give an example of this rigidity. An affine real algebraic variety $X$ is rigid if, for each affine irreducible real algebraic variety $Z$, the set of all nonconstant regular morphisms from...

We show that, if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points, then it is conjugate to a real meromorphic function by a suitable projective automorphism of the image.