A Bijective Proof for a Theorem of Ehrhart

Sam, Steven V

doi:10.4169/193009709x460813

Cited by 27 publications

(9 citation statements)

References 14 publications

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“…Then P I = i∈I P {i} for all ∅ = I ⊆ [r]. As in [15] we observe that (k + d)∆ d−1 = P ∅ = i∈[d] P {i} for all k ≥ 0. Therefore, by inclusion-exclusion,…”

Section: Characteristic Functions and Valuationsmentioning

confidence: 53%

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Linear recursions for integer point transforms

Jochemko¹

2019

Preprint

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“…Then P I = i∈I P {i} for all ∅ = I ⊆ [r]. As in [15] we observe that (k + d)∆ d−1 = P ∅ = i∈[d] P {i} for all k ≥ 0. Therefore, by inclusion-exclusion,…”

Section: Characteristic Functions and Valuationsmentioning

confidence: 53%

“…Therefore the order of the minimal polynomial of the sequence is dim P as was demonstrated in [15] and is thus in general smaller than |V(P )|.…”

Section: A Multivariate Recursionmentioning

confidence: 83%

Linear recursions for integer point transforms

Jochemko¹

2019

Preprint

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“…A detailed proof following the instructions of [6,Section 3.7] is documented in Barco [5]. (See Sam [90] for an alternative approach.) Note that for any polytope ∆, all the faces in its boundary ∂∆ must be convex.…”

Section: Normal Affine Semigroup Rings and Ehrhart Theorymentioning

confidence: 99%

The Shape of Hilbert-Kunz Functions

Chan¹

2021

Preprint

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We discuss Hilbert-Kunz function from when it was originally defined to its recent developments. A brief history of Hilbert-Kunz theory is first recounted. Then we review several techniques involved in the study of Hilbert-Kunz functions by presenting some illustrative proofs without going into details of the technicalities.The second part of this article focuses on the Hilbert-Kunz function of an affine normal semigroup ring and relates it to Ehrhart quasipolynomials. We pay extra attention to its periodic behavior and discuss how the cellular decomposition constructed by Bruns and Gubeladze fits into the computation of the functions. The closed forms of the Hilbert-Kunz function of some examples are presented. The discussion in this part highlights the close relationship between Hilbert-Kunz function and Ehrhart theory.

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“…Lately there has been a lot of literature on various topics generated by the Ehrhart polynomial and we cite just a few papers and books: [1], [2], [3], [4], [5], [6], [7], [8], [9], [11], [28], [29], and [31].…”

Section: Introductionmentioning

confidence: 99%

Ehrhart's polynomial for equilateral triangles in $\mathbb Z^3$

Ionascu

2011

Preprint

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In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in Z 3 . The polynomial takes a relatively simple form in terms of the coordinates of the vertices of the polytope and it depends heavily on the value d and its divisors, where d = a 2 +b 2 +c 2 3 and (a, b, c) (gcd(a, b, c) = 1) is a vector with integer coordinates normal to the plane containing the triangle.

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A Bijective Proof for a Theorem of Ehrhart

Cited by 27 publications

References 14 publications

Linear recursions for integer point transforms

Linear recursions for integer point transforms

The Shape of Hilbert-Kunz Functions

Ehrhart's polynomial for equilateral triangles in $\mathbb Z^3$

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