  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.
