Abstract
Let E be a ring of entire functions on \(\mathbb{C}^\user1{N} \) with the operation of pointwise multiplication, and let f1,...,fm be a set of nonzero elements in E. The ideal E(f1,...,fm) in E with generators f1,...,fm is said to be generating if E(f1,...,fm) = E. The generating ideals in rings of entire functions on \(\mathbb{C}^\user1{N} \) determined by the growth of their maximum moduli are characterized in terms of the distribution of the zero sets of their generators. Under the additional condition of rapid variation of the weight sequences determining the ring, criteria for generating ideals are established; they are stated in terms of d(z) ≔ max 1 ≤ j ≤mdj(z), where dj(z) is the distance from a point \(\user1{z} \in \mathbb{C}^\user1{N} \) to the zero set of fj for 1 ≤ j ≤m. It is shown that, in rings of entire functions of finite or minimal type with respect to a given order, a similar characterization (i.e., in terms of d(z)) cannot be given.
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call