We study Diophantine equations of type $$f(x)=g(y)$$f(x)=g(y), where both f and g have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials $$(f_n)_n$$(fn)n are such that $$f_n$$fn satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K of characteristic zero, and at least two distinct critical points and all distinct critical values, then the monodromy group of f is a doubly transitive permutation group. In particular, f cannot be represented as a composition of lower degree polynomials. Several authors have studied monodromy groups of polynomials with some similar properties. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if $$f(x)=g(h(x))$$f(x)=g(h(x)) with $$g, h\in K[x]$$g,h∈K[x] and $$\deg g>1$$degg>1, then either $$\deg h\le 2$$degh≤2, or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points.

中文翻译：

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分离变量中的丢番图方程

我们研究了 $$f(x)=g(y)$$f(x)=g(y) 类型的丢番图方程，其中 f 和 g 至少有两个不同的临界点（导数的根）并且相等的临界点值最多两个不同的临界点。多项式的各种经典族 $$(f_n)_n$$(fn)n 使得 $$f_n$$fn 满足所有 n 的这些假设。我们的结果涵盖并概括了文献中关于此类方程积分解的有限性的几个结果。为此，我们分析了此类多项式的单项群的性质。我们证明，如果 f 在特征为零的域 K 中具有系数，并且至少有两个不同的临界点和所有不同的临界值，那么 f 的单向群是一个双传递置换群。特别是，f 不能表示为低次多项式的组合。几位作者研究了具有一些相似性质的多项式的单项群。我们进一步证明，如果 f 至少有两个不同的临界点并且至多有两个临界值相等，并且如果 $$f(x)=g(h(x))$$f(x)=g(h (x)) 用 $$g, h\in K[x]$$g,h∈K[x] 和 $$\deg g>1$$$degg>1，则 $$\deg h\le 2 $$degh≤2，或者 f 是特殊类型。特别是在后一种情况下，f 没有三个简单的临界点，也没有五个不同的临界点。