"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.
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