We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute bounds on the maximal order type.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-1-2,
author = {Matthias Aschenbrenner and Wai Yan Pong},
title = {Orderings of monomial ideals},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {27-74},
zbl = {1049.06001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-1-2}
}
Matthias Aschenbrenner; Wai Yan Pong. Orderings of monomial ideals. Fundamenta Mathematicae, Tome 184 (2004) pp. 27-74. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-1-2/