Compressed ideals

Mermin, Jeff

doi:10.1112/blms/bdm095

Bulletin of the London Mathematical Society

2008

DOI: 10.1112/blms/bdm095

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Compressed ideals

Jeff Mermin

Abstract: Compressed ideals have been used by Macaulay and others to study Hilbert functions and lex ideals in a polynomial ring. We formalize the theory of compression, classify the compressed ideals, and provide new proofs for theorems of Macaulay, Kruskal-Katona, and Bigatti, Hulett, and Pardue.

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Cited by 7 publications

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Betti numbers of lex ideals over some Macaulay-Lex rings

Mermin

Murai

2009

J Algebr Comb

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. . . , x n ] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R = A/M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl-Füredi-Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus-M ideals have maximal Betti numbers when A/M is Macaulay-Lex.

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Betti numbers of lex ideals over some Macaulay-Lex rings

Mermin

Murai

2009

J Algebr Comb

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Hilbert functions of colored quotient rings and a generalization of the Clements–Lindström theorem

Chong

2014

J Algebr Comb

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The Lex-Plus-Powers Conjecture holds for pure powers

Mermin

Murai

2011

Advances in Mathematics

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Compressed ideals

Abstract: Compressed ideals have been used by Macaulay and others to study Hilbert functions and lex ideals in a polynomial ring. We formalize the theory of compression, classify the compressed ideals, and provide new proofs for theorems of Macaulay, Kruskal-Katona, and Bigatti, Hulett, and Pardue.

Cited by 7 publications

References 18 publications

Betti numbers of lex ideals over some Macaulay-Lex rings

Betti numbers of lex ideals over some Macaulay-Lex rings

Hilbert functions of colored quotient rings and a generalization of the Clements–Lindström theorem

The Lex-Plus-Powers Conjecture holds for pure powers

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