Properties of Entire Functions Over Polynomial Rings via Gröbner Bases

Abstract

In this paper it is shown that the extension ideals of polynomial prime and primary ideals in the corresponding ring of entire functions remain prime or primary, respectively. Moreover, we will prove that a primary decomposition of a polynomial ideal can be extended componentwise to a primary decomposition of the extended ideal. In order to show this we first prove the flatness of the ring of entire functions over the corresponding polynomial ring by use of Grobner basis techniques. As an application we give an elementary proof of a generalization of Hilbert's Nullstellensatz for entire functions (cf. [10]).