This paper considers some properties of meromorphic solutions of the nonlinear di¤erence equationwhere Pðz; f ðzÞÞ and Qðz; f ðzÞÞ are polynomials in f having rational coe‰cients and no common roots.

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This paper studies the properties of companion matrices of primitive polynomials over finite field and discusses the recognition of a primitive polynomial by companion matrix. It is showed-is a prime number then the recognition of a primitive polynomial over finite field Fp of degree n can reduce to the recognition of a primitive element. When 2, p 2 this can reduce the complexity of the recognition of a primitive polynomial.

In this paper, we mainly investigate properties of finite order transcendental meromorphic solutions of difference Painlevé equations. If f is a finite order transcendental meromorphic solution of difference Painlevé equations, then we get some estimates of the order and the exponent of convergence of poles of f (z), where f (z) = f (z + 1) -f (z). MSC: 30D35; 39B12

In this paper, we mainly investigate some properties of the transcendental meromorphic solution f (z) for the difference Riccati equation f (z + 1) = p(z)f (z)+q(z) f (z)+s(z) . We obtain some estimates of the exponents of the convergence of the zeros and poles of f (z) and the difference f (z) = f (z + 1) -f (z). MSC: 30D35; 39B12