Title:  Stanley Decompositions and Involutive Bases 

Author: Apel, Joachim 

Affiliation: MSRI Berkeley, California

Abstract: In 1982 Richard P. Stanley conjectured that a graded module M over a
graded k-algebra R can be decomposed in a particular way in a direct sum of
finitely many free modules over suitable subalgebras of R.  Besides homogeneity
conditions the most important restriction which such a decomposition has to
satisfy is that the subalgebras must have at least dimension depth(M). For
monomial ideals I of a polynomial ring R such Stanley decompositions of M1=I
and M2=R/I are strongly related to decompositions of the set of derivatives
occurring in the Riquier/Janet method for solving systems of PDE's.  We will
show how general involutive bases may be applied in order to prove some
particular cases of Stanley's conjecture and to provide algorithms for the
computation of Stanley decompositions.