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 ρ(f)=r→ +∞log(log(|f|(r)) log r. When ρ(f)≠ 0, +∞, one can define the type of growth as σ(f)=r→ +∞log(|f|(r)) rρ(f). But here, we can also define the cotype of growth as ψ(f)=r→ +∞q(f,r) rρ(f) where $q(f,r)$ is the number of zeros of f in the disk of center $0$ and radius r. Then we have ρ(f)σ(f)≤ ψ(f)≤ eρ(f)σ(f). Moreover, if ψ or σ are veritable limits, then ρ(f)σ(f)= ψ(f) and this relation is conjectured in the general case. Many other properties are examined concerning ρ(f),\ σ(f),\ ψ(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.