Pure O-Sequences: Known Results, Applications, and Open Problems

Migliore, Juan; Nagel, Uwe; Zanello, Fabrizio

doi:10.1007/978-1-4614-5292-8_16

“…We don't have any formula to describe the Hilbert function of I (m) X C . For m = 1 Proposition 4.1 implies that the h-vector of X C is a pure O-sequence (see [39] for all the terminology and background on this topic). This is not always true if m > 1.…”

Section: Corollary 46mentioning

confidence: 99%

Steiner systems and configurations of points

Ballico¹,

Favacchio²,

Guardo³

et al. 2020

Des. Codes Cryptogr.

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The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.

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“…3 on p. 508 by Huh, which I have already described. On the other hand, looking at the unimodality of sequences arising from commutative algebra is now a large area of research (see, e.g., [36], and the references cited there). Regarding the conjectures, both statements in the first part of Conj.…”

Section: Log-concave and Unimodal Sequences In Algebra Combinatoricsmentioning

confidence: 99%

Stanley’s work on unimodality

Brenti¹

2016

The Mathematical Legacy of Richard P. Stanley

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“…We have β XC . For m = 1 Proposition 4.1 implies that the h-vector of X C is a pure O-sequence (see [43] for all the terminology and background on this topic). This is not always true if m > 1.…”

mentioning

confidence: 99%

Steiner systems and configurations of points

Ballico¹,

Favacchio²,

Guardo³

et al. 2020

Preprint

0

View full text Add to dashboard Cite

The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.

show abstract

Pure O-Sequences: Known Results, Applications, and Open Problems

Abstract: Abstract. This note presents a discussion of the algebraic and combinatorial aspects of the theory of pure O-sequences. Various instances where pure O-sequences appear are described. Several open problems that deserve further investigation are also presented.

Cited by 5 publications

References 75 publications

Steiner systems and configurations of points

Steiner systems and configurations of points

Stanley’s work on unimodality

Steiner systems and configurations of points

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