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 [X1, · · · , Xn] and the power series ring F [X][[Y ]] = F [X1, · · · , Xn][[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 λ on their Y -degree. These rings arise naturally in studying p-adic analytic variation of zeta functions over finite fields. Future research into this area may study more complicated subrings in which Y = (Y1, · · · , Ym) has more than one variable, and for which there are multiple degree functions, λ1, · · · , λm. Another direction of study would be to generalize these results to k-affinoid algebras.