论文标题
理性动力系统，$ s $ - 单位和$ d $ -Finite Power系列
Rational dynamical systems, $S$-units, and $D$-finite power series
论文作者
论文摘要
令$ k $为特征零的代数封闭字段，让$ g $成为$ k $的乘法组的有限生成的子组。我们考虑表格$ a_n：= f（φ^n（x_0））$的$ k $值序列，其中$φ\ colon x \ to x $ to x $ and $ f \ colon x \ to \ mathbb {p}^1 $是在$ k $上定义了$ k $和$ x_0 $的$ x $的$ x $ y是$ x $ a的point and point and toper的$ c an $ f $。数字理论和代数组合制剂的许多经典序列都属于这个动态框架，我们表明，$ n $的集合是$ n $ in g $中的$ a_n \ in g $中的算术进展有限的结合，以及一组Banach密度零。此外，我们表明，如果每$ n $ and $ n $ and $ x $ in G $ in g $ in g $ in g $ in g $ in n $ x $ in $ x $ in $ x $ in $ x $都是不可估的，那么有一个多重蛋糕$ \ mathbb {g} _mm^d $ and maps $^d $和maps $ cumbb $ \ mathbb; \ Mathbb {g} _m^d $和$ g：\ Mathbb {g} _m^d \ to \ Mathbb {g} _m $，以至于$ a_n = g \ circ Circin^n（y）$ for Some $ y \ in \ Mathbb in \ MathBB {g} _mm^d $。然后，我们使用这些事实获得有关$ d $ finite功率系列系数的结果。
Let $K$ be an algebraically closed field of characteristic zero and let $G$ be a finitely generated subgroup of the multiplicative group of $K$. We consider $K$-valued sequences of the form $a_n:=f(φ^n(x_0))$, where $φ\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $φ$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and $X$ is irreducible and the $φ$ orbit of $x$ is Zariski dense in $X$ then there are a multiplicative torus $\mathbb{G}_m^d$ and maps $Ψ:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ Ψ^n(y)$ for some $y\in \mathbb{G}_m^d$. We then obtain results about the coefficients of $D$-finite power series using these facts.
