Applications of the $p$-adic Nevanlinna theory to functional equations

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

“…Proposition 4.4 (Bautabaa and Escassut [3,4] Proof. Clearly, f 0 P 0 ðf Þ ¼ g 0 Q 0 ðgÞ: Since P 0 Q 0 a0; either f 0 ¼ g 0 ¼ 0; or f 0 g 0 a0: Suppose f 0 ¼ g 0 ¼ 0: By Proposition 4.4 there exist f 1 ; g 1 AMðKÞ such that ðf 1 Þ w ¼ f ; ðg 1 Þ w ¼ g: Then, we have P 1 ðf 1 Þ ¼ Q 1 ðg 1 Þ and therefore we are led to the same problem with f 1 and g 1 : Thus, by induction, after t similar operations we finally Theorem 4.6.…”

“…Many applications of the Nevanlinna's values distribution Theory were made in padic analysis as in complex analysis, to study uniqueness problems [15] and the parametrization of curves or functional equations [3][4][5]. In the present paper, we are examining whether a meromorphic function may admit two different decompositions of the form P3f and Q3g; where P; Q are two nonlinear polynomials.…”

“…(2) In a field of characteristic 0; the above functionsZðr; f Þ andÑðr; f Þ are just those also denoted by % Zðr; f Þ and % Nðr; f Þ; respectively [3][4][5].…”

Value Distribution Theory Related to Number Theory

“…hyperelliptic) curve. Hence one can apply Theorem 5.43 with m = 2, deg(B) = 0, deg(A) = s = 3 and m = 2, deg(B) = 0, deg(A) = s ≥ 4 in Corollary 5.44, respectively (see [35]). According to the proof of Theorem 4.60, we can obtain the following result: where l = c∈S µ 0 P (c).…”

“…Thus we have G 2 (M(C)) = 4. (4.6.10) in the class C[z], C(z), and A(C), respectively, which are due to Montel [275], Jategaonkar [197], Yang [438], and Gross [130], [131] (or see [33], [35], [132], [176] ). Since the rational functions (cf.…”