Asymptotic Hilbert polynomials and limiting shapes

Dumnicki, Marcin; Szpond, Justyna; Tutaj-Gasińska, Halszka

doi:10.1016/j.jpaa.2015.02.026

Cited by 7 publications

(32 citation statements)

References 13 publications

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“…In the present note, we specialize the lines so that they intersect. It has been discussed in [12,Example 20] and generalized in [9, Example 5] that in case of intersecting lines there is a correction term in the coefficients of Λ s . We have Λ 2 (t) = t 3 − 6t + 4 for a pair of skew lines, whereas for a pair of intersecting lines we have Λ 2,1 (t) = t 3 − 6t + 6.…”

Section: The Limits Of the Methodsmentioning

confidence: 99%

Lower Bounds for Waldschmidt Constants of Generic Lines in $${\mathbb {P}}^3$$ P 3 and a Chudnovsky-Type Theorem

et al. 2019

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The Waldschmidt constant α(I) of a radical ideal I in the coordinate ring of P N measures (asymptotically) the degree of a hypersurface passing through the set defined by I in P N. Nagata's approach to the 14th Hilbert Problem was based on computing such constant for the set of points in P 2. Since then, these constants drew much attention, but still there are no methods to compute them (except for trivial cases). Therefore, the research focuses on looking for accurate bounds for α(I). In the paper, we deal with α(s), the Waldschmidt constant for s very general lines in P 3. We prove that α(s) ≥ √ 2s − 1 holds for all s, whereas the much stronger bound α(s) ≥ √ 2.5s holds for all s but s = 4, 7 and 10. We also provide an algorithm which gives even better bounds for α(s), very close to the known upper bounds, which are conjecturally equal to α(s) for s large enough.

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Section: The Limits Of the Methodsmentioning

confidence: 99%

Lower Bounds for Waldschmidt Constants of Generic Lines in $${\mathbb {P}}^3$$ P 3 and a Chudnovsky-Type Theorem

et al. 2019

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“…In the paper [7] the authors define the so-called asymptotic Hilbert function and asymptotic Hilbert polynomial. Namely, let K be an algebraically closed field of characteristic zero.…”

Section: Preliminariesmentioning

confidence: 99%

“…It is worth observing that we do not know, so far, any example of a homogeneous ideal that is not LBSR. The list of ideals which are proved to be LBSR may be found, e.g., in [7]. Now we recall the definitions of the asymptotic Hilbert function and the asymptotic Hilbert polynomial of an ideal I.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recently the asymptotic invariants of ideals have aroused great interest and have been studied by many researchers, see, for example, [8], [9], [6], [7], [19], [18], [20], and others. One of these asymptotic invariants is the so-called Waldschmidt constant of an ideal I ⊂ K[P n ] denoted by α(I), see, for example, [3], [4], [2].…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper we generalize this result and give a method to compute an upper bound of α(I) for a wide class of ideals (namely, radical ideals with linearly bounded regularity of symbolic powers, see Preliminaries for the definitions). To find this bound we use aHP I (t), the socalled asymptotic Hilbert polynomial of I, defined in [7]. The bound is given by a ASYMPTOTIC HILBERT POLYNOMIAL AND BOUND FOR WALDSCHMIDT CONSTANTS 9 root of a suitable derivative of the polynomial Λ I (t) := t n n!…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Hilbert polynomial and a bound for Waldschmidt constants

Dumnicki¹,

Farnik²,

Tutaj-Gasińska³

2016

ERA-MS

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Asymptotic invariants constructed from generic initial ideals

Malara

2023

Mathematische Nachrichten

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Generic initial ideals (gins for short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they have been of interest in commutative algebra and indirectly in algebraic geometry. Recently, Mayes in a series of articles associated with gins of graded families of ideals geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes, and some asymptotic invariants of homogeneous ideals which are associated with the gins, for example, asymptotic regularity, Waldschmidt constant and some new invariants, which seem relevant from geometric point of view.

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Asymptotic Hilbert polynomials and limiting shapes

Cited by 7 publications

References 13 publications

Lower Bounds for Waldschmidt Constants of Generic Lines in $${\mathbb {P}}^3$$ P 3 and a Chudnovsky-Type Theorem

Lower Bounds for Waldschmidt Constants of Generic Lines in $${\mathbb {P}}^3$$ P 3 and a Chudnovsky-Type Theorem

Asymptotic Hilbert polynomial and a bound for Waldschmidt constants

Asymptotic invariants constructed from generic initial ideals

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