Gradings of Polynomial Rings

These brief notes have their origin in a conversation I had with Abhyankar during a pleasant walk along the Wabash a few years ago. While I tried to explain some ideas on torus actions on affine spaces Abhyankar, in that inimitable way he has to concentrate one’s mind, feigned innocence as to such fancy notions, but was quite happy to listen when I proposed talking about gradings of polynomial rings instead. The two topics cover exactly the same ground from two different points of view, equally valid and valuable. Torus actions also provide an uncomplicated introduction to the larger subject of reductive group actions. I gave my conference talk hoping that such an introduction in the readily understood language of gradings would be appreciated by some of the non-experts in the rather diverse audience Abhyankar’s birthday was bound to bring together. This is a slightly expanded version of my talk. It is entirely expository in nature and not meant for the specialists.