Let A, B, C be some nonconstant polynomials with coefficients in a field K of characteristic zero. We prove that the polynomial Pillai equation \(A^x-B^y=C\) has at most one solution in positive integers x, y except for three explicitly described triplets \((A,B,C) \in K[t]^3\) when it has exactly two solutions in \((x,y) \in \mathbb {N}^2\).

中文翻译：

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多项式的Pillai方程

令A，  B，  C为某些非常数多项式，其系数在特征K的字段K中为零。我们证明多项式Pillai方程\（A ^ xB ^ y = C \）最多有一个正整数x，y的解， 除了三个明确描述的三元组\（（A，B，C）\ in K [t] ^ 3 \）在\（（x，y）\ in \ mathbb {N} ^ 2 \）中有两个解时。