# Seminar in Real & Complex Geometry

## Degree-like functions in complex and real algebraic geometry

Abstract

 "Degree-like functions" give a natural tool for construction of projective compactifications of affine algebraic varieties. The first part of the talk will be an introduction to this theory. The second part of the talk will be about an application of this theory in real algebraic geometry. More precisely, given a semi-algebraic subset S of R^n (which goes to infinity) and a positive integer d, let B_d(S) be the set of all (real) polynomials f in n variables such that there is a polynomial h of degree d such that f^2 less or equal to h on S. Real algebraic geometers study the algebra B(S) formed by the direct sum of all B_d(S) in order to understand the how polynomials grow on S. Now consider the function \delta on the ring of real polynomials (in n variables) which assigns to a polynomial f the smallest integer d such thatf belongs to B_d(S). Then \delta defines a degree-like function on the ring of polynomials (over the complex numbers!), and the study of the corresponding compactifications of C^n immediately gives answers to some questions about finite generation of B(S). The reference for the first part of the talk is arXiv:1012.0835. The second part is an ongoing work with Tim Netzer.