1. CJM 2010 (vol 62 pp. 1228)
||Valuations for Matroid Polytope Subdivisions|
We prove that the ranks of the subsets and the activities of the bases
of a matroid define valuations for the subdivisions of a matroid
polytope into smaller matroid polytopes.
Categories:05B35, 52B40, 52B45, 52C22
2. CJM 2009 (vol 61 pp. 888)
||Face Ring Multiplicity via CM-Connectivity Sequences |
The multiplicity conjecture of Herzog, Huneke, and Srinivasan
is verified for the face rings of the following classes of
simplicial complexes: matroid complexes, complexes of dimension
one and two,
and Gorenstein complexes of dimension at most four.
The lower bound part of this conjecture is also established for the
face rings of all doubly Cohen--Macaulay complexes whose 1-skeleton's
connectivity does not exceed the codimension plus one as well as for
all $(d-1)$-dimensional $d$-Cohen--Macaulay complexes.
The main ingredient of the proofs is a new interpretation
of the minimal shifts in the resolution of the face ring
$\field[\Delta]$ via the Cohen--Macaulay connectivity of the
skeletons of $\Delta$.
Categories:13F55, 52B05;, 13H15;, 13D02;, 05B35