Abstract:Given a `degree like' function $\delta$, i.e. a map of the coordinate ring $A$ of an affine variety $X$ to integers such that $\delta(f+g) \leq \max\{\delta(f), \delta(g)\}$ (with strict inequality implying $\delta(f) = \delta(g)$) and $\delta(fg) \leq \delta(f) + \delta(g)$, one can associate to it a projective completion of $X^\delta$ of $X$. We show that the ideal $I_\infty$ of the hypersurface of `points at infinity' is radical iff $\delta$ is the maximum of finitely many semidegrees (i.e. degree like functions $\delta'$ such that $\delta'(fg) = \delta'(f) + \delta'(g)$ for all $f,g$), which are then in a 1-1 correspondence with the prime components of $I_{\infty}$. Given a projective completion $Z$ of $X$ determined by a degree like function $\delta$, we define a ?normalized? degree like function $\bar \delta$ which is a maximum of finitely many semidegrees. When $X$ is normal, the corresponding completion is isomorphic to the normalization of $Z$. The construction of $\bar \delta$ from $\delta$ generalizes the construction of the normal toric completion $X_P$ of $(C*)^n$ determined by a convex integral polytope $P$ from the toric variety determined by an arbitrary finite subset $S$ of integral points in $P$ whose convex hull is $P$. I will describe this construction and the properties of toric completions these are so far known to preserve.