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Newton Complementary Duals of $f$-Ideals

Part of: Algebraic combinatorics Arithmetic rings and other special rings

Published online by Cambridge University Press:  15 October 2018

Samuel Budd  and
Adam Van Tuyl
Samuel Budd
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4L8 Email: [email protected]@[email protected]
Adam Van Tuyl
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4L8 Email: [email protected]@[email protected]
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Abstract

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A square-free monomial ideal $I$ of $k[x_{1},\ldots ,x_{n}]$ is said to be an $f$-ideal if the facet complex and non-face complex associated with $I$ have the same $f$-vector. We show that $I$ is an $f$-ideal if and only if its Newton complementary dual $\widehat{I}$ is also an $f$-ideal. Because of this duality, previous results about some classes of $f$-ideals can be extended to a much larger class of $f$-ideals. An interesting by-product of our work is an alternative formulation of the Kruskal–Katona theorem for $f$-vectors of simplicial complexes.

Keywords

f-idealsfacet idealStanley–Reisner correspondencef-vectorsNewton complementary dualKruskal–Katona

MSC classification

Primary: 05E45: Combinatorial aspects of simplicial complexes
Secondary: 05E40: Combinatorial aspects of commutative algebra 13F55: Stanley-Reisner face rings; simplicial complexes
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 231 - 241
Copyright
© Canadian Mathematical Society 2018 

Footnotes

1

Current address for Budd: 2061 Oliver Road, Thunder Bay, ON, P7G 1P7

Parts of this paper appeared in the first author’s MSc project [4]. The second author’s research was supported in part by NSERC Discovery Grant 2014-03898.

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