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We establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P(f)\prod_{i=1}^k(f^{(i)})^{n_i}$ and $P(g)\prod_{i=1}^k(g^{(i)})^{n_i}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp.