The functional equation P(f)=Q(g) in a p-adic field

Abstract

Let K be a complete ultrametric algebraically closed field of characteristic p: Let P; Q be in K½x with P 0 Q 0 not identically 0: Consider two different functions f ; g analytic or meromorphic inside a disk jx À ajor (resp. in all K), satisfying Pðf Þ ¼ QðgÞ: By applying the Nevanlinna's values distribution Theory in characteristic p; we give sufficient conditions on the zeros of P 0 ; Q 0 to assure that both f ; g are ''bounded'' in the disk (resp. are constant). If pa2 and degðPÞ ¼ 4; we examine the particular case when Q ¼ lP (lAK) and we derive several sets of conditions characterizing the existence of two distinct functions f ; g meromorphic in K such that Pðf Þ ¼ lPðgÞ: r

Value Distribution Theory Related to Number Theory

“…Theorem 5.45 improves a result in [96]. Note that when p > 0, if there exist two non-constant meromorphic functions f and g in κ satisfying (5.10.9), then f and g have the same ramification index s and p s √ P p s f = p s Q ( p s √ g) .…”