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1
Content available remote Polynomial mappings with small degree
EN
Let Xn be an affine variety of dimension n and Yn be a quasi-projective variety of the same dimension. We prove that for a quasi-finite polynomial mapping f : Xn → Yn ,every non-empty component of the set Yn\f(Xn) is closed and it has dimension greater or equal to (…), where (…) is a geometric degree of f. Moreover, we prove that generally, if (…) is any polynomial mapping, then either every non-empty component of the set (…) is of dimension (…) or f contracts a subvariety of dimension (…).
2
Content available remote Multi-sorted logic and logical geometry : some problems
EN
The paper has a form of a survey on basics of logical geometry and consists of three parts. It is focused on the relationship between many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appears. A list of problems which naturally arise in this field is another objective of the paper.
3
Content available remote Abstract Barycentric Algebras
EN
This paper presents a new approach to the study of (real) barycentric algebras, in particular convex subsets of real affine spaces. Barycentric algebras are cast in the setting of two-sorted algebras. The real unit interval indexing the set of basic operations of a barycentric algebra is replaced by an LP-algebra, the algebra of ukasiewicz Product Logic. This allows one to define barycentric algebras abstractly, independently of the choice of the unit real interval. It reveals an unexpected connection between barycentric algebras and (fuzzy) logic. The new class of abstract barycentric algebras incorporates barycentric algebras over any linearly ordered field, the B-sets of G. M. Bergman, and E. G. Manes' if-then-else algebras over Boolean algebras.
4
EN
We generalize the known construction of "parabein model" of the affine plane geometry to arbitrary finite dimension and study some transformations characteristic for the obtained structures.
5
Content available remote A proof of the projective Desargues axiom in the Desarguesian affine plane
EN
We give a short proof that the projective Desargues axiom is valid in the Desarguesian affine planes.
6
Content available remote Affine geometry of spine spaces
EN
The parallelity relation and the group of dilatations in the geometry of spine spaces are investigated. Fundamental theorems of affine geometry are proved and the analytical representation of dilatations is given.
7
Content available remote The set of fixed points of an algebraic group
EN
Let G be an algebraic group, which acts effectively on an affine space C[sup n]. Assume that a hypersurface W [is a subset of a set] C[sup n] is contained in the set of fixed point of G. Then W is a C-uniruled variety, i.e. there exists a n - 1-dimensional affine cylinder R x C and a dominant, generically-finite polynomial mapping phi : R x C --> W.
8
Content available remote Injections into affine hypersurfaces
EN
Let X be a smooth affine variety of dimension n > 2. Assume that the group H[sub 1](X,Z) is a torsion group and that [chi](X) = 1. Let Y be a projectively smooth affine hypersurface Y [is a subset of] C[sup n+1] of degree d > 1, which is smooth at infinity. Then there is no injective polynomial mapping f : X --> Y. This contradicts a result of Peretz [5].
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