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A new Clunie type theorem for difference polynomials. (English) Zbl 1213.39005
Summary: It is shown that if $w(z)$ is a finite-order meromorphic solution of the equation $$H(z,w)P(z,w)= Q(z,w),$$ where $P(z,w)= P(z,w(z),w(z+c_1),\dots,w(z+c_n))$, $c_1,\dots,c_n\in\Bbb C$, is a homogeneous difference polynomial with meromorphic coefficients, and $H(z,w)=H(z,w(z))$ and $Q(z,w)= Q(z,w(z))$ are polynomials in $w(z)$ with meromorphic coefficients having no common factors such that $$\max\{\deg_w(H),\deg_w(Q)-\deg_w(P)\}> \min\{\deg_w(P), \text{ord}_0(Q)\}- \text{ord}_0(P),$$ where $\text{ord}_0(P)$ denotes the order of zero of $P(z,x_0,x_1,\dots,x_n)$ at $x_0=0$ with respect to the variable $x_0$, then the Nevanlinna counting function $N(r,w)$ satisfies $N(r,w)\ne S(r,w)$. This implies that $w(z)$ has a relatively large number of poles. For a smaller class of equations, a stronger assertion $N(r,w)= T(r,w)+ S(r,w)$ is obtained, which means that the pole density of $w(z)$ is essentially as high as the growth of $w(z)$ allows. As an application, a simple necessary and sufficient condition is given in terms of the value distribution pattern of the solution, which can be used as a tool in ruling out the possible existence of special finite-order Riccati solutions within a large class of difference equations containing several known difference equations considered to be of Painlevé type.

##### MSC:
 39A10 Additive difference equations 39A12 Discrete version of topics in analysis 30D35 Distribution of values (one complex variable); Nevanlinna theory
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