Marcin Bilski

Approximation of analytic sets along Nash subvarieties

Abstract Let X be an analytic subset of pure dimension n of an open set U ⊂ Cm and let E be a Nash subset of U such that E ⊂ X.Then for every a ∈ E there is an open neighborhood V of a in U and a sequence {Xv} of complex Nash subsets of V of pure dimension n converging to X ∩ V in the sense of holomorphic chains such that the following hold for every v ∈ N: E ∩ V ⊂ Xv and the multiplicity of Xv at x equals the multiplicity of X at x for every x in a dense open subset of E ⊂ V.

On Nash approximation of complex analytic sets in Runge domains

Abstract We prove that every complex analytic set X in a Runge domain Ω can be approximated by Nash sets on any relatively compact subdomain Ω 0 of Ω. Moreover, for every Nash subset Y of Ω with Y ⊂ X , the approximating sets can be chosen so that they contain Y ∩ Ω 0 . As a consequence, we derive a necessary and sufficient condition for a complex analytic set X to admit a Nash approximation which coincides with X along its arbitrary given subset.

Approximation of holomorphic maps from Runge domains to affine algebraic varieties

The aim of this paper is to give an analytic proof of the theorem on algebraic approximations of holomorphic maps from Runge domains to affine algebraic varieties.

Generative complexity in semigroup varieties

Abstract We analyze the problem of generative complexity for varieties of semigroups. We focus our attention on varieties generated by a finite semigroup. A variety is said to have polynomial generative complexity if and only, if the number of k -generated semigroups it contains is bounded from above by a polynomial in k . We fully characterize the class of finite semigroups which generate varieties with polynomial complexity. It turns out that only semigroups of very special shape have this property. These are semigroups with zero multiplication or semigroups which are the product of a left-zero semigroup, a right-zero semigroup and an Abelian group. Moreover a characterization of finite semigroups generating varieties with linear complexity is given.

HIGHER ORDER APPROXIMATION OF ANALYTIC SETS BY TOPOLOGICALLY EQUIVALENT ALGEBRAIC SETS

It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.