Georg Regensburger: "Symbolic computation for linear operators with matrix coefficients"
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For symbolic computations with systems of linear functional equations, like (integro-)differential equations or boundary problems, we need an algebraic framework that enables effective computations in corresponding rings of operators. To represent and compute with concrete linear systems usually matrices of operators with scalar coefficients are used. For statements about whole families of linear functional systems, however, symbolic methods that directly work with operators having undetermined matrix coefficients of generic size need to be used. In the talk, we discuss a construction of integro-differential operators over an arbitrary integro-differential ring. Allowing noncommutative coefficients, we can treat functional systems of generic size with this approach. We outline how we can find and prove using computer algebra all consequences of the fundamental theorem of calculus in differential rings. Our general approach is based on tensor reduction systems and allows us to find and compute with normal forms for certain rings of linear operators. Normal forms can also be used to prove identities as well as to solve operator equations by ansatz. We illustrate our approach with examples for some classes of linear functional systems. The talk is based on joint work with Jamal Hossein Poor and Clemens Raab.