FACTORIAL AND NOETHERIAN SUBRINGS OF POWER SERIES RINGS

Let F be a field. We show that certain subrings contained between the polynomial ring F[X] = F[X(1), ... , X(n)] and the power series ring F[X][[Y]] = F[X(1), ... , X(n)][[Y]] have Weierstrass Factorization, which allows us to deduce both unique factorization and the Noetherian property. These intermediate subrings are obtained from elements of F[X][[Y]] by bounding their total X-degree above by a positive real-valued monotonic up function lambda on their Y-degree. These rings arise naturally in studying the p-adic analytic variation of zeta functions over finite fields. Future research into this area may study more complicated subrings in which Y = (Y(1), ... , Y(m)) has more than one variable, and for which there are multiple degree functions, lambda(1), ... , lambda(m). Another direction of study would be to generalize these results to k-affinoid algebras.