The talk will be about a method for explicit construction of
compactifications of $C^n$ with some desired properties at infinity. The
main tool for the construction is a global variant of "key polynomial" of a
valuation (key polynomial was introduced by MacLane and is a basic tool in
valuation theory). The original motivation was applications to the "affine
Bezout problem" of finding number of solutions in $C^n$ of a given system of
polynomials; however, the explicitness of the construction makes it
applicable to other problems, e.g. computation of cohomologies of line
bundles, or determination of topology at infinity, and a big part of these
problems boils down to combinatorics in almost the same spirit of
(projective) toric varieties. In this talk I will focus on the dimension two
case, where things are relatively more transparent. The general case will
(hopefully) be the topic of a future talk.
