On the functional equation Af 2 + Bg 2 = 1 on the field of complex p-adic numbers

Abstract

For a fixed prime p, let Cp denote the complex p-adic numbers. For polynomials A, B ∈ Cp[x] we consider decompositions A(x)f 2 (x)+B(x)g 2 (x) = 1 of entire functions f, g on Cp and try to improve an impossibility result due to A. Boutabaa concerning transcendental f, g. We also provide an independent proof of a p-adic diophantic statement due to D. N. Clark, which is an important ingredient of Boutabaa's method.