Daniel Robertz
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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