Minimal Polygons with Fixed Lattice Width

Cools, Filip; Lemmens, Alexander

doi:10.1007/s00026-019-00431-0

Cited by 5 publications

(5 citation statements)

References 9 publications

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“…. ∧ P p,i ⊗ Q i + terms not containing P in the ∧ part (12) where all P j,i 's are in ∆ and Q i ∈ ∆. We claim that in fact Q i ∈ ∆ , i.e.…”

Section: Pruning Off Vertices Without Changing the Lattice Widthmentioning

confidence: 89%

“…But one can do better: in a spin-off paper [12] devoted to minimal polygons, the second and the fourth author show that if ∆ is a minimal lattice polygon with lw(∆) ≤ d then…”

Section: Pruning Off Vertices Without Changing the Lattice Widthmentioning

confidence: 99%

“…Sharpness in the cases where N ∆ ≤ 32 is obtained by explicit verification, based on the data from [7] and using the algorithm described in Section 8; this covers more than half a million (unimodular equivalence classes of) small lattice polygons. The statement involving lw(∆) ≤ 6 relies on this exhaustive verification, along with the classification of inclusion-minimal lattice polygons having a given lattice width, which is elaborated in [12].…”

Section: Introductionmentioning

confidence: 99%

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Computing graded Betti tables of toric surfaces

Castryck

Cools

Demeyer

et al. 2019

Trans. Amer. Math. Soc.

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We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the linear strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology, and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25, depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface ν 6 (P 2 ) ⊆ P 27 in characteristic 40 009. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface ν d (P 2 ).

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“…. ∧ P p,i ⊗ Q i + terms not containing P in the ∧ part (12) where all P j,i 's are in ∆ and Q i ∈ ∆. We claim that in fact Q i ∈ ∆ , i.e.…”

Section: Pruning Off Vertices Without Changing the Lattice Widthmentioning

confidence: 89%

“…But one can do better: in a spin-off paper [12] devoted to minimal polygons, the second and the fourth author show that if ∆ is a minimal lattice polygon with lw(∆) ≤ d then…”

Section: Pruning Off Vertices Without Changing the Lattice Widthmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing graded Betti tables of toric surfaces

Castryck

Cools

Demeyer

et al. 2019

Trans. Amer. Math. Soc.

Self Cite

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“…This classification provides an alternative proof of Corollary 17. Note that a classification of inclusion-minimal lattice polygons P with fixed lattice width w(P ) was provided in [7].…”

Section: Introductionmentioning

confidence: 99%

Bounds on Area Involving Lattice Size

Soprunova

2023

Electron. J. Combin.

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The lattice size of a lattice polygon $P$ is a combinatorial invariant of $P$ that was recently introduced in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve. In this paper, we establish sharp lower bounds on the area of plane convex bodies $P\subset\mathbb{R}^2$ that involve the lattice size of $P$. In particular, we improve bounds given by Arnold, and Bárány and Pach. We also provide a classification of minimal lattice polygons $P\subset\mathbb{R}^2$ of fixed lattice size ${\operatorname{ls_\square}}(P)$.

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“…This classification provides an alternative proof of the above bound. Note that a classification of inclusion-minimal lattice polygons P with fixed lattice width w(P ) was provided in [5].…”

Section: Introductionmentioning

confidence: 99%

Lattice Size and Area of Plane Convex Bodies

Soprunova¹

2021

Preprint

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The lattice size of a lattice polygon P was introduced and studied by Schicho, and by Castryck and Cools in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve. In this paper we establish sharp lower bounds on the area of plane convex bodies P ⊂ R 2 that involve the lattice size of P . In particular, we improve bounds established by Arnold, and Bárány and Pach. We also provide a classification of minimal lattice polygons P ⊂ R 2 of fixed lattice size ls (P ).

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Minimal Polygons with Fixed Lattice Width

Abstract: We classify the unimodular equivalence classes of inclusion-minimal polygons with a certain fixed lattice width. As a corollary, we find a sharp upper bound on the number of lattice points of these minimal polygons.

Cited by 5 publications

References 9 publications

Computing graded Betti tables of toric surfaces

Computing graded Betti tables of toric surfaces

Bounds on Area Involving Lattice Size

Lattice Size and Area of Plane Convex Bodies

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