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

✍ Scribed by Alain Escassut; Chung-Chun Yang

• Book ID: 104024388
• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 292 KB
• Volume: 105
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2003.11.005

✦ Synopsis

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: