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Factorization and resultants of partial differential operators. (English) Zbl 1275.35013

Summary: Comparatively little is known about commutative rings of partial differential operators, while in the ordinary case, concrete examples and an algebraic(-geometric) structure can be algorithmically determined for large classes. In this note, by the calculation of the partial \(\mu\)-shifted differential resultant which we defined in a previous paper, we produce algebraic equations of spectral surfaces for commutative rings in two variables, and Darboux transformations of Airy-type operators that correspond to morphisms of surfaces. There are, however, many elementary differential-algebraic statements that we only observe experimentally, thus we offer open questions which seem to us quite significant in differential algebra, and access to Mathematica code to enable further experimentation.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35-04 Software, source code, etc. for problems pertaining to partial differential equations

Software:

Mathematica
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