On the functional equation Af^2+Bg^2=1 in a p-adic field of characteristic zero

Abstract

For a fixed prime $p$, let ${\mathbb{C}}_p$ denote the complex $p$-adic numbers. For polynomials $A,B\in {\mathbb{C}}_p[x]$ we consider decompositions $A(x)f^2(x)+B(x)g^2(x)=1$ of entire functions $f,g$ on ${\mathbb{C}}_p$ 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.

arXiv comment10 pages