We introduce a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposed by a direct sum of integral domains (using idempotent elements) that enjoys certain technical features and that the coefficients of the difference equation are not degenerated. Using this mechanism we can reduce the problem to find solutions in a ring (with zero-divisors) to search solutions in several copies of integral domains. Utilizing existing solvers in this integral domain setting, we obtain a general solver where the components of the linear difference equations and the solutions can be taken from difference rings that are built e.g., by $R\Pi\Sigma$-extensions over $\Pi\Sigma$-fields. This class of difference rings contains, e.g., nested sums and products, products over roots of unity and nested sums defined over such objects.
翻译:我们引入了一般削减战略, 使一个人能够在差异环中寻找参数线性差异方程式的解决方案。 在这里, 我们假设环本身可以被具有某些技术特点的集成域( 使用一元元素) 的直接组合( 使用一元元素) 分解, 并且差异方程式的系数不会下降。 使用这个机制, 我们可以减少问题, 在一个环( 带零divisors ) 中找到解决方案, 在多个集成域中寻找解决方案 。 利用这个集成域设置中的现有解决方案, 我们获得一个普通求解器, 将线性差异方程式的成分和解决方案从例如 $R\ Pi\ Sigma$- expeension 所建的“ $\ Pi\ Sigma$- fields” 的分界圈中提取。 这个差异项圈包括, 例如, 嵌套总和产品、 嵌套件的产品, 以及这些对象定义的嵌套数 。