On projective completions of affine varieties determined by 'degree-like' functions

We study projective completions of affine algebraic varieties which are given by filtrations, or equivalently, 'degree like functions' on their rings of regular functions. For a quasifinite polynomial map P (i.e. with all fibers finite) of affine varieties, we prove that there are completions of the source that do not add points at infinity for P (i.e. in the intersection of completions of the hypersurfaces corresponding to a generic fiber and determined by the component functions of P). Moreover we show that there are 'finite type' completions with the latter property, determined by the maximum of a finite number of 'semidegrees', i.e. maps of the ring of regular functions excluding zero, into integers, which send products into sums and sums into maximas (with a possible exception when the summands have the same semidegree). We characterize the latter type completions as the ones for which the ideal of the 'hypersurface at infinity' is radical. Moreover, we establish a one-to-one correspondence between the collection of minimal associated primes of the latter ideal and the unique minimal collection of semidegrees needed to define the corresponding degree like function. We also prove an 'affine Bezout type' theorem for quasifinite polynomial maps P which admit semidegrees such that corresponding completions do not add points at infinity for P.