二次和其它非递归高阶差分方程的因子分解和降阶
Factorizations and Reductions of Order in Quadratic and other Non-recursive Higher Order Difference Equations
摘要：
高阶差分方程通常可以在任意非空集S中定义为：\[ f_{n}(x_{n},x_{n-1},...,x_{n-k})=g_{n}(x_{n},x_{n-1},...,x_{n-k}) \]其中$f_{n},g_{n} :S^{k+1}\rightarrow S$是给定的$n=1,2,... $的函数，并且$k$是正整数。我们提出的条件，意味着上述方程可以分解成一个等价的对低阶差分方程使用可能的形式对称性（减少变量的变化）。这些结果推广了群上递归差分方程的共轭分解。我们应用这一理论的一些获得新的因式分解结果的重要类二次差分方程的代数领域。我们还讨论了二次方程解的存在性的非平凡问题。
英文摘要：
A higher order difference equation may be generally defined in an arbitrary nonempty set S as: \[ f_{n}(x_{n},x_{n-1},...,x_{n-k})=g_{n}(x_{n},x_{n-1},...,x_{n-k}) \] where $f_{n},g_{n} :S^{k+1}\rightarrow S$ are given functions for $n=1,2,... $ and $k$ is a positive integer. We present conditions that imply the above equation can be factored into an equivalent pair of lower order difference equations using possible form symmetries (order-reducing changes of variables). These results extend and generalize semiconjugate factorizations of recursive difference equations on groups. We apply some of this theory to obtain new factorization results for the important class of quadratic difference equations on algebraic fields. We also discuss the nontrivial issue of the existence of solutions for quadratic equations.
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