Noetherian property of subrings of power series rings

Abstract

Let R be a commutative ring with unit. We study subrings R[X; Y, lambda] of R[X][[Y]] = R[X-1, ..., X][[Y-1, ..., Y-m]], where lambda is a nonnegative real-valued increasing function. These rings R[X; Y, lambda] are obtained from elements of R[X][[Y]] by bounding their total X-degree above by lambda on their Y-degree. Such rings naturally arise from studying p-adic analytic variation of zeta functions over finite fields. Under certain conditions, Wan and Davis showed that if R is Noetherian, then so is R[X; Y, lambda]. In this article, we give a necessary and sufficient condition for R[X; Y, lambda] to be Noetherian when Y has more than one variable and I grows at least as fast as linear. It turns out that the ring R[X; Y, lambda] is not Noetherian for a quite large class of functions I including functions that were asked about by Wan.