论文标题
Burchnall-Chaundy多项式用于基质ODOS和PICARD-VESSIOT理论
Burchnall-Chaundy polynomials for matrix ODOs and Picard-Vessiot Theory
论文作者
论文摘要
Burchnall和Chaundy表明，如果两个ODOS $ p $，$ Q $带有分析系数通勤，则存在一个多项式$ f（λ，μ）$，具有复杂系数，例如$ f（p，q）= 0 $，称为BC-PolyNomial。可以使用ODOS的差分结果来计算此多项式。在这项工作中，我们将此结果扩展到矩阵普通微分运算符Modos。矩阵有差异字段$ k $的条目，其常数$ c $的字段是代数关闭的，特征为零。我们限于订单One Oner Onerator $ p $的情况，并具有可逆的领先系数。定义了一种新的差分消除工具，即矩阵差异结果。它用于计算一对通勤Modos的BC多项式$ F $，并证明具有恒定系数。该结果为光谱问题提供了必要和充分的条件，$ py =λy\，\ qy =μy$具有解决方案。来自差异代数和Picard-Vessiot理论的技术使我们能够在Modos $ C [P，Q] $的交换环与不可减至的代数曲线环的有限产物之间明确描述同构。
Burchnall and Chaundy showed that if two ODOs $P$, $Q$ with analytic coefficients commute there exists a polynomial $f(λ,μ)$ with complex coefficients such that $f(P,Q)=0$, called the BC-polynomial. This polynomial can be computed using the differential resultant for ODOs. In this work we extend this result to matrix ordinary differential operators, MODOs. Matrices have entries in a differential field $K$, whose field of constants $C$ is algebraically closed and of zero characteristic. We restrict to the case of order one operators $P$, with invertible leading coefficient. A new differential elimination tool is defined, the matrix differential resultant. It is used to compute the BC-polynomial $f$ of a pair of commuting MODOs and proved to have constant coefficients. This resultant provides the necessary and sufficient condition for the spectral problem $PY=λY \ , \ QY=μY$ to have a solution. Techniques from differential algebra and Picard-Vessiot theory allow us to describe explicitly isomorphisms between commutative rings of MODOs $C[P,Q]$ and a finite product of rings of irreducible algebraic curves.
