标题： 关于复合多项式$f(g(x)) =f(x)h^m(x)$的方程 显示英文标题

标题： On the equation $f(g(x)) =f(x)h^m(x)$ for composite polynomials

摘要： 在本文中，我们求解方程$f(g(x))=f(x)h^m(x)$，其中$f(x)$、$g(x)$和$h(x)$是系数在任意域$K$中的未知多项式，$f(x)$是非常数且可分的，$\deg g \geq 2$，多项式$g(x)$在$K[x]$中具有非零导数$g'(x) \ne 0$，整数$m \geq 2$不被域$K$的特征整除。 我们证明当$\deg f \geq 3$时，这个方程没有解。 如果$\deg f = 2$，我们证明$m = 2$并以切比雪夫多项式的形式明确给出所有解。 此类多项式$f(x)$,$g(x)$,$h(x)$在系数属于$\Q$或$\Z$的情况下，其不定方程应用是在 Cassaign 等人的猜想背景下进行研究的。 对卢瓦尔函数在点$\lambda$的值，以及$f(r)$，$r \in \Q$处的值 显示英文摘要

摘要： In this paper we solve the equation $f(g(x))=f(x)h^m(x)$ where $f(x)$, $g(x)$ and $h(x)$ are unknown polynomials with coefficients in an arbitrary field $K$, $f(x)$ is non-constant and separable, $\deg g \geq 2$, the polynomial $g(x)$ has non-zero derivative $g'(x) \ne 0$ in $K[x]$ and the integer $m \geq 2$ is not divisible by the characteristic of the field $K$. We prove that this equation has no solutions if $\deg f \geq 3$. If $\deg f = 2$, we prove that $m = 2$ and give all solutions explicitly in terms of Chebyshev polynomials. The diophantine applications for such polynomials $f(x)$, $g(x)$, $h(x)$ with coefficients in $\Q$ or $\Z$ are considered in the context of the conjecture of Cassaign et. al on the values of Louiville's $\lambda$ function at points $f(r)$, $r \in \Q$.