The starting point of this note is the observation that the local condition used in the notion of a Hilbert-symbol equivalence and a quaternion-symbol equivalence — once it is expressed in terms of the Witt invariant — admits a natural generalisation. In this paper we show that for global function fields as well as the formally real function fields over a real closed field all the resulting equivalences coincide.

The most accurate determinateness criteria for the multivariate moment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing measure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and...

We give a state-of-the-art survey of investigations concerning multivariate polynomial inequalities. A satisfactory theory of such inequalities has been developed due to applications of both the Gabrielov-Hironaka-Łojasiewicz subanalytic geometry and pluripotential methods based on the complex Monge-Ampère operator. Such an approach permits one to study various inequalities for polynomials restricted not only to nice (nonpluripolar) compact subsets of ℝⁿ or ℂⁿ but also their versions for pieces...

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.

We give an estimate of Siciak’s extremal function for compact subsets of algebraic varieties in ${ℂ}^n$ (resp. ${ℝ}^n$). As an application we obtain Bernstein-Walsh and tangential Markov type inequalities for (the traces of) polynomials on algebraic sets.