We prove a characteristic $p$ version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. In characteristic $p$, the Thue–Siegel–Dyson–Roth theorem is false, so the proof requires new techniques from those used by Silverman. The problem is largely that isotriviality can arise in subtle ways, and we define and compare three different definitions of isotriviality for maps, sets, and curves. Using results of Favre and Rivera-Letelier on the structure of Julia sets, we prove that if $\phi$ is a nonisotrivial rational function and $\beta$ is not exceptional for $\phi$, then ${\phi }^{-n}(\beta )$ is a nonisotrivial set for all sufficiently large $n$; we then apply diophantine results of Voloch and Wang that apply for all nonisotrivial sets. When $\phi$ is a polynomial, we use the nonisotriviality of ${\phi }^{-n}(\beta )$ for large $n$ along with a partial converse to a result of Grothendieck in descent theory to deduce the nonisotriviality of the curve $y^{\ell }={\phi }^n(x)-\beta$ for large $n$ and small primes $\ell \ne p$ whenever $\beta$ is not postcritical; this enables us to prove stronger results on Zsigmondy sets. We provide some applications of these results, including a finite index theorem for arboreal representations coming from quadratic polynomials over function fields of odd characteristic.

我们证明一个特性$p$西尔弗曼定理关于数域上轨道积分点的版本，并在此设置下建立多项式的本原素除数定理。在特点上 $p$，Thue-Siegel-Dyson-Roth 定理是错误的，因此证明需要 Silverman 使用的新技术。问题主要在于等平凡性可以以微妙的方式出现，我们定义并比较了地图、集合和曲线的等平凡性的三种不同定义。利用 Favre 和 Rivera-Letelier 在 Julia 集结构上的结果，我们证明如果$\phi$是一个非等平凡有理函数并且$\beta$并不例外$\phi$， 然后${\phi }^{-n}（\beta ）$是所有足够大的非等平凡集$n$; 然后我们应用适用于所有非等平凡集的 Voloch 和 Wang 的丢番图结果。什么时候$\phi$是一个多项式，我们使用非等平凡性${\phi }^{-n}（\beta ）$对于大$n$以及对格洛腾迪克下降理论结果的部分逆推，推导出曲线的非等平凡性$y^{\ell }={\phi }^n（X）-\beta$对于大$n$和小素数$\ell \ne p$每当$\beta$不是后批判的；这使我们能够在 Zsigmondy 集上证明更强的结果。我们提供了这些结果的一些应用，包括来自奇数特征函数域上的二次多项式的树栖表示的有限索引定理。