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1

Polynomial mappings with small degree

Jelonek Z.

Demonstratio Mathematica

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2015

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Vol. 48, nr 2

242--249

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

Multi-sorted logic and logical geometry : some problems

Plotkin B., Plotkin E.

Demonstratio Mathematica

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2015

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Vol. 48, nr 4

578--619

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

A proof of the projective Desargues axiom in the Desarguesian affine plane

Prażmowska M.

Demonstratio Mathematica

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2004

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Vol. 37, nr 4

921--924

EN

We give a short proof that the projective Desargues axiom is valid in the Desarguesian affine planes.

4

The set of fixed points of an algebraic group

Jelonek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 1

69-73

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.

5

Note about the set S[f] for a polynomial mapping f : [C^2 --> C^2]

Jelonek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 1

67-72

EN

The aim of this paper is to give a simple method of computing the set S[f] of points at which a generically-finite polynomial mapping f : [C^2 --> C^2] is not proper.

6

Injections into affine hypersurfaces

Jelonek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 1

63-67

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].