§ Ring of power series with infinite positive and negative terms
If we allow a ring with elements for all , for notation's
sake, let's call it . Unfortunately, this is a badly behaved ring.
Define . See that , since
multiplying by shifts powers by 1. Since we are summing over all of ,
is an isomorphism. Rearranging gives . If we want our ring
to be an integral domain, we are forced to accept that . In the Barvinok
theory of polyhedral point counting, we accept that and exploit this
in our theory.