Uniqueness and Value Sharing of Differential Polynomials over a Non-Archimedean Field

Let K be an algebraically closed field of characteristic zero, complete for a nontrivial non-Archimedean absolute value. Let a,b∈ K\0\ and let $$n,k,p$$ be positive integers. We study non-constant functions f and g whose differential polynomials fⁿ+a(f⁽ᵏ⁾)ᵖ and gⁿ+a(g⁽ᵏ⁾)ᵖ share the value b CM . For entire functions, if n≥ p+2 and (n,p)=1 , then $$f=g$$ unless f and g are proportional polynomials of degree at most k . For meromorphic functions, if n≥ 2p(k+1)+8 , then f and g satisfy one of three explicit algebraic relations. These results extend earlier work of Tuan and Quang and, for $$p=1$$ , improve their bounds on n .