A note on the van der Waerden complex

  • Becky Hooper
  • Adam Van Tuyl

Abstract

Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when these pure simplicial complexes are vertex decomposable or not Cohen-Macaulay. As a corollary, we classify the van der Waerden complexes that are shellable.

References

Eagon, J. A. and Reiner, V., Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), no. 3, 265–275. https://doi.org/10.1016/S0022-4049(97)00097-2

Ehrenborg, R., Govindaiah, L., Park, P. S., and Readdy, M., The van der Waerden complex, J. Number Theory 172 (2017), 287–300. https://doi.org/10.1016/j.jnt.2016.08.012

Ene, V. and Herzog, J., Gröbner bases in commutative algebra, Graduate Studies in Mathematics, vol. 130, American Mathematical Society, Providence, RI, 2012.

Herzog, J. and Hibi, T., Monomial ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-106-6

Hooper, B., Shellability of the van der waerden complex, M.Sc.\
project, McMaster University, 2017.

Provan, J. S. and Billera, L. J., Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), no. 4, 576–594. https://doi.org/10.1287/moor.5.4.576

Villarreal, R. H., Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238, Marcel Dekker, Inc., New York, 2001.
Published
2019-06-17
How to Cite
Hooper, B., & Van Tuyl, A. (2019). A note on the van der Waerden complex. MATHEMATICA SCANDINAVICA, 124(2), 179-187. https://doi.org/10.7146/math.scand.a-111923
Section
Articles