非阿基米德环上隐式线性微分方程的Cramer法则

A. Goncharuk
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引用次数: 0

摘要

研究了一类系数为特征为零的形式幂级数环上的线性非齐次$m$-阶微分方程。这个方程在这个环中有无穷多个解——对应柯西问题的每个初始条件都有一个解。这些解可以用微分方程理论的经典方法求得。我们假设方程的系数和非齐次性系数属于某个积分域K。我们要从这个积分域中寻找带系数的幂级数形式的解。经典理论的方法不允许我们找出是否存在一个初始条件对应于系数K的解,也不允许找到这个初始条件。为了解决这个问题,我们采用了U. Broggi提出的方法。这种方法允许以某种特殊级数的形式找到线性非齐次微分方程的形式解。在前面的文章中,对于一类非阿基米德值环$K$,给出了解存在唯一性的充分条件。如果这些条件成立,则考虑用Broggi方法得到的形式幂级数。它的系数是在考虑的非阿基米德拓扑中收敛的级数的和。证明了这个级数是方程K[[x]]$的解。注意,这个形式幂级数环上的方程可以被认为是一个关于未知形式幂级数系数的无限线性方程组。本文证明了该系统可以用类似于Cramer方法的方法来求解,其中无限矩阵的行列式是在非阿基米德拓扑中某些有限行列式的极限。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Cramer's rule for implicit linear differential equations over a non-Archimedean ring
We consider a linear nonhomogeneous $m$-th order differential equation in a ring of formal power series with coefficients from some field of characteristic zero. This equation has infinite many solutions in this ring -- one for each initial condition of the corresponding Cauchy problem. These solutions can be found using classical methods of differential equation theory. Let us suppose the coefficients of the equation and the coefficients of nonhomogeneity belong to some integral domain $K$. We are looking for a solution in the form of a formal power series with coefficients from this integral domain. The methods of classical theory do not allow us to find out whether there exists an initial condition that corresponds to the solution of the coefficients from $K$ and do not allow find this initial condition. To solve this problem, we use the method proposed by U. Broggi. This method allows to find a formal solution of the linear nonhomogeneous differential equation in the form of some special series. In previous articles, sufficient conditions for the existence and uniqueness of a solution were found for a certain class of rings $K$ with a non-Archimedean valuation. If these conditions hold, the formal power series obtained using the Broggi’s method is considered. Its coefficients are the sums of series that converge in the non-Archimedean topology considered. It is shown that this series is the solution from $K[[x]]$ of our equation. Note that this equation over a ring of formal power series can be considered as an infinite linear system of equations with respect to the coefficients of unknown formal power series. In this article it is proved that this system can be solved by some analogue of Cramer's method, in which the determinants of infinite matrices are found as limits of some finite determinants in the non-Archimedean topology.
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