Meromorphic Mapping Unicity Theorem And The Derivative Of The Entire Function

This paper focuses on the unicity theorem for the meromorphic mappingsand the total derivative of entire functions.The first is a unicity theorem with truncated multiplicities of meromorphicmappings in several complex variables for moving targets. Let f,g : C^m→CPⁿ be two non-constant meromorphic mappings , a₁,…,a_q be small with respect to f , and in general position such that (f, a_j) (?) 0 and (g,a_j) (?) 0, 1≤j≤q. If (a) min{v_(f,aj),d} = min{v_(f,aj),d} (1≤j≤q), where 1≤d≤n; (b) dim{z∈C^m|(f,a_i)(z) = (f,a_j)(z) = 0}≤m - 2 (1≤i≤j≤q)> (c) f(z) = g(z), z∈∪_j=1^q{z∈C^m|(f,a_j)(z) = 0}, we have, (i)If q≥4n² + 2n + 3 - 2d,then f≡g. (ii)If f and g are linearly non-degenerate over R({a_j}_j=1^q) and q≥2n² + 4n + 3 - 2d, then f≡g. This improved Ru's result in [2].The second is a unicity theorem for the total derivative. Let f be an entire function on Cⁿ , p be a polynomial on Cⁿ such that p(0) = 0 . If D^k f and pf CM share a finite a≠0 , andδ(0, f)＞1/2 , then D^k f = pf. This generalized Liu and Gu' s theorem in [3] to the entire function of several complex variables.