I recently came across a tidy little result I proved when at Imperial College a while ago and that, as it was peripheral to what I was primarily doing, has not seen the light of day since. Now that I have this new medium to output periphery into, I thought it might be a good idea to air it. It also provided a good opportunity to review complete intersections, which are a class of geometrically motivated rings with many appealing computational properties. I couldn’t find much content at the time (nor now) regarding the roots of Hilbert series, but there are some blatant connections between them and the structure of the rings from which they arise. Here is my note with some exposition and then said tidy little result.