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On solutions of linear functional systems and factorization of modules over Laurent-Ore algebras
| Content Provider | Semantic Scholar |
|---|---|
| Copyright Year | 2005 |
| Abstract | A Laurent-Ore algebra L over a field F is a mathematical abstraction of common properties of linear partial differential and difference operators. A linear (partial) functional system is of the form A(z) = 0 where A is a matrix over L and z is a vector of unknowns. Typically, it is a system consisting of linear partial differential, shift and q-shift operators, or any mixture thereof. We associate to a linear functional system A(z) = 0 an L-module MA, which is called the module of formal solutions. For our purpose, the dimension of an L-module is defined to be the dimension of the module as a vector space over F . A system A(z) = 0 is said to be ∂-finite if MA has finite dimension. A Picard-Vessiot extension for a ∂-finite system A(z) = 0 is a ring containing “all” solutions of A(z) = 0. We prove the existence of Picard-Vessiot extensions for all ∂-finite linear functional systems and show that the dimension of the solution space of a ∂-finite system equals the dimension of its module of formal solutions. The Grobner basis techniques for left ideals in Ore algebras are extended to left submodules over Laurent-Ore algebras. This extension enables us to determine whether a linear functional system is ∂-finite. We present an algorithm for finding all submodules of an L-module with finite dimension. This algorithm allows us to find all “subsystems” whose solution spaces are contained in that of a given ∂-finite linear functional system. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.mmrc.iss.ac.cn/~mwu/Thesis/Wu-thesis.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
