Order, type and cotype of growth for p-adic entire functions A survey with additional properties

International audienceLet IK be a complete ultrametric algebraically closed field and let A(IK) be the IK-algebra of entires functions on IK. For an f ∈ A(IK), similarly to complex analysis, one can define the order of growth as ρ(f) = lim sup r→+∞ log(log(|f |(r)) log r. When ρ(f) = 0, +∞, one can define the type of growth as σ(f) = lim sup r→+∞ log(|f |(r)) r ρ(f). But here, we can also define the cotype of growth as ψ(f) = lim sup r→+∞ q(f, r) r ρ(f) where q(f, r) is the number of zeros of f in the disk of center 0 and radius r. Many properties described here were first given in the Houston Journal, but new inequalities linking the order, type and cotype are given in this paper: we show that 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. Several other properties are examined concerning ρ, σ, ψ for f and f. Particularly, we show that if an entire function f has finite order, then f f 2 takes every value infinitely many times