Order, Type and Cotype of Growth for p-adic Entire Function

Let be a complete ultrametric algebraically closed field and let A () be the -algebra of entires functions on. For an f A (), similarly to complex analysis, one can define the order of growth as (f) =ₑ + ( (|₅| (ₑ) ) ₑ. When (f) 0, +, one can define the type of growth as (f) =ₑ + (|₅| (ₑ) ) ₑ^{ (₅) }. But here, we can also define the cotype of growth as (f) =ₑ +ₐ (₅, ₑ) ₑ^{ (₅) } where q (f, r) is the number of zeros of f in the disk of center 0 and radius r. Then we have (f) (f) (f) e (f) (f). Moreover, if or are veritable limits, then (f) (f) = (f) and this relation is conjectured in the general case. Many other properties are examined concerning (f), \ (f), \ (f). Particularly, we prove that if an entire function f has finite order, then f' f² takes every value infinitely many times and applications are shown to branched values of meromorphic functions. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.