RWTH Aachen University

Title: Janet's algorithm for modules over polynomial rings

Abstract: This talk gives an introduction to Janet bases.

Originally developed for the algebraic analysis of systems

of partial differential equations in the beginning of the

20th century, the algorithm by Maurice Janet is today an

efficient alternative for Buchberger's algorithm to compute

Gr{\"o}bner bases of modules over polynomial rings.

In this talk we give a modern description of Janet's

algorithm and explain nice combinatorial properties of

the resulting Janet bases: separation of the variables into

multiplicative and non-multiplicative ones for each Janet

basis element allows to read off vector space bases for

both the submodule and the residue class module. As a

consequence, the Hilbert series and polynomial of a (graded)

module as well as a free resolution are easily obtained

from the Janet basis.

If time admits, some modifications of Janet's algorithm

will be addressed which allow to work with polynomial rings

over the integers instead of a field resp. generalize the

algorithm to certain classes of non-commutative polynomial

rings.

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