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

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