Zbigniew Jelonek

A solution of the problem of van den Essen and Shpilrain

Abstract In this paper we solve a problem of van den Essen and Shpilrain about endomorphisms, which preserve coordinate polynomials. More precisely, we show that every endomorphism of C [x 1 ,…,x n ] taking any coordinate polynomial to a coordinate one is an automorphism.

IRREDUCIBLE IDENTITY SETS FOR POLYNOMIAL AUTOMORPHISMS

In this paper we study polynomial automorphisms of some algebraic affine varieties. The main tool of our investigations is the notion of the uniruled set. The paper is divided into five parts (sections 1–5 respectively). The first one is an introduction. In the second one we set up notations, definitions and terminology. In section 3 we study the set of fixed points of a polynomial automorphism of Cn. The nature of this set is not well-known even in the case when an automorphism has a finite order (see [10]). Let us recall the notions which we shall use in the sequel. Let X ⊂ Cn be an algebraic set. We say that X is an identity set for polynomial automorphisms of Cn if every two polynomial automorphisms that coincide on X must be equal. X is a set determining polynomial automorphisms of Cn if the same holds under the weaker assumption that these two automorphisms transforms X onto the same set. In [13] Wang and McKay proved that two intersecting lines are an identity set in C2, later in [14] they generalized this result on n hyperplanes in general position in Cn (n ≥ 2). In our paper [8] we generalized these results and we proved that the union of every n hypersurfaces in general position in Cn is an identity set in Cn (n ≥ 2).

The set of fixed points of a unipotent group

Abstract Let K be an algebraically closed field. Let G be a non-trivial connected unipotent group, which acts effectively on an affine variety X. Then every non-empty component R of the set of fixed points of G is a K-uniruled variety, i.e., there exist an affine cylinder W × K and a dominant, generically-finite polynomial mapping ϕ : W × K → R . We show also that if an arbitrary infinite algebraic group G acts effectively on K n and the set of fixed points contains a hypersurface H, then this hypersurface is K-uniruled.

Affine automorphisms that are isometries

Abstract Let L 1 ,…,L m ⊂ R n be k -dimensional affine subspaces, which are not co-conical ( n > k >0). We show that if F: R n → R n is an affine automorphism, which preserves k -dimensional volumes on all subspaces L 1 ,…, L m , then F is an isometry.

Bifurcation locus and branches at infinity of a polynomial f:\mathbb {C}^2\rightarrow \mathbb {C}

We show that the number of bifurcation points at infinity of a polynomial function f : C → C is at most the number of branches at infinity of a generic fiber of f and that this upper bound can be diminished by one in certain cases.

Maximal sets of linearly independent vectors in a free module over a commutative ring

The following theorem is proved: If R is a noetherian ring, M is a free R-module of rank n, and {v1,…, vs} is the maximal set of linearly independent vectors, then always s=n. An example is also given of a commutative ring R for which the above theorem is false for every n⩾2.

Quantitative properties of the non-properness set of a polynomial map

Let

Testing sets for properness of polynomial mappings

X\subset\mathbb{C}^n

The set of points at which a polynomial map is not proper

be an affine variety covered by polynomial curves, and let

The extension of regular and rational embeddings

f:X\rightarrow\mathbb{C}^m