Group Gradings on Polynomial Algebras

Let k be a field. We consider gradings on a polynomial algebra k[X1,…, Xn] by an arbitrary abelian group G, such that the indeterminates are homogeneous elements of nontrivial degree. We classify the isomorphism types of such gradings, and we count them in the case where G is finite. We present some examples of good gradings and find a minimal set of generators of the subalgebra of elements of trivial degree.