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arXiv:1107.4862v1 [math.CO] 25 Jul 2011

EHRHART POLYNOMIALS OF INTEGRAL SIMPLICES

WITH PRIME VOLUMES

AKIHIRO HIGASHITANI

Abstract. For an integral convex polytope P ⊂ RNof dimension d, we call δ(P) =

(δ0,δ1,...,δd) its δ-vector and vol(P) =?d

we will establish the new equalities and inequalities on δ-vectors for integral simplices

whose normalized volumes are prime. Moreover, by using those, we will classify all the

possible δ-vectors of integral simplices with normalized volume 5 and 7.

i=0δi its normalized volume. In this paper,

Introduction

One of the most fascinating problems on enumerative combinatorics is to characterize

the δ-vectors of integral convex polytopes.

Let P ⊂ RNbe an integral convex polytope of dimension d, which is a convex polytope

any of whose vertices has integer coordinates. Let ∂P denote the boundary of P. Given

a positive integer n, we define

i(P,n) = |nP ∩ ZN|,i∗(P,n) = |n(P \ ∂P) ∩ ZN|,

where nP = {nα : α ∈ P} and |X| is the cardinality of a finite set X. The enumerative

function i(P,n) is called the Ehrhart polynomial of P, which was studied originally in the

work of Ehrhart [1]. The Ehrhart polynomial has the following fundamental properties:

• i(P,n) is a polynomial in n of degree d. (Thus, in particular, i(P,n) can be defined

for every integer n.)

• i(P,0) = 1.

• (loi de r´ eciprocit´ e) i∗(P,n) = (−1)di(P,−n) for every integer n > 0.

We refer the reader to [2, Part II] and [7, pp. 235–241] for the introduction to the theory

of Ehrhart polynomials.

We define the sequence δ0,δ1,δ2,... of integers by the formula

?∞

n=0

(1 − λ)d+1

?

i(P,n)λn

?

=

∞

?

i=0

δiλi. (1)

Then, from a fundamental result on generating function ([7, Corollary 4.3.1]), we know

that δi= 0 for every i > d. We call the integer sequence

δ(P) = (δ0,δ1,...,δd),

2000 Mathematics Subject Classification: Primary 52B20; Secondary 52B12.

Keywords: Integral simplex, Ehrhart polynomial, δ-vector.

The author is supported by JSPS Research Fellowship for Young Scientists.

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which appears in (1), the δ-vector of P. In addition, by the reciprocity law, one has

∞

?

n=1

i∗(P,n)λn=

?d

i=0δd−iλi+1

(1 − λ)d+1

.

The δ-vector has the following fundamental properties:

• δ0= 1 and δ1= |P ∩ ZN| − (d + 1).

• δd= |(P \ ∂P) ∩ ZN|. Hence, we have δ1≥ δd.

• Each δiis nonnegative ([8]).

• If (P \ ∂P) ∩ ZNis nonempty, then one has δ1≤ δifor every 1 ≤ i ≤ d − 1 ([3]).

• When d = N, the leading coefficient (?d

?d

Recently, the δ-vectors of integral convex polytopes have been studied intensively. For

example, see [6], [10] and [11].

There are two well-known inequalities on δ-vectors. Let s = max{i : δi?= 0}. One is

i=0δi)/d! of i(P,n) is equal to the usual

volume of P ([7, Proposition 4.6.30]). In general, the positive integer vol(P) =

i=0δiis said to be the normalized volume of P.

δ0+ δ1+ ··· + δi≤ δs+ δs−1+ ··· + δs−i,0 ≤ i ≤ ⌊s/2⌋,(2)

which is proved in Stanley [9], and another one is

δd+ δd−1+ ··· + δd−i≤ δ1+ δ2+ ··· + δi+ δi+1,0 ≤ i ≤ ⌊(d − 1)/2⌋,(3)

which appears in Hibi [3, Remark (1.4)].

When?d

pletely ([4, Theorem 5.1]) by (2) and (3) together with an additional condition. Further-

more, by the proofs of [5, Theorem 0.1] and [4, Theorem 5.1], we know that all the possible

δ-vectors can be realized as the δ-vectors of integral simplices when?d

further classifications of the δ-vectors with?d

straints on δ-vectors for integral simplices whose normalized volumes are prime numbers.

The following theorem is our main result of this paper.

i=0δi≤ 3, the above inequalities (2) and (3) characterize the possible δ-vectors

completely ([5]). Moreover, when?d

i=0δi= 4, the possible δ-vectors are determined com-

i=0δi≤ 4. However,

unfortunately, it is not true when?d

δ-vectors of integral simplices. In this paper, in particular, we establish some new con-

i=0δi= 5. (See [4, Remark 5.2].) Therefore, for the

i=0δi≥ 5, it is natural to investigate the

Theorem 0.1. Let P be an integral simplex of dimension d and δ(P) = (δ0,δ1,...,δd)

its δ-vector. Suppose that?d

Then,

i=0δi= p is an odd prime number. Let i1,...,ip−1 be the

positive integers such that?d

(a) one has

i=0δiti= 1 + ti1+ ··· + tip−1with 1 ≤ i1≤ ··· ≤ ip−1≤ d.

i1+ ip−1= i2+ ip−2= ··· = i(p−1)/2+ i(p+1)/2≤ d + 1;

(b) one has

ik+ iℓ≥ ik+ℓ for 1 ≤ k ≤ ℓ ≤ p − 1 with k + ℓ ≤ p − 1.

We prove Theorem 0.1 in Section 1 via the languages of elementary group theory.

As an application of Theorem 0.1, we give a complete characterization of the possible

δ-vectors of integral simplices when?d

i=0δi= 5 and 7.

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[2] T. Hibi, “Algebraic Combinatorics on Convex Polytopes,” Carslaw Publications, Glebe NSW, Aus-

tralia, 1992.

[3] T. Hibi, A lower bound theorem for Ehrhart polynomials of convex polytopes, Adv. in Math. 105

(1994), 162 – 165.

[4] T. Hibi, A. Higashitani and N. Li, Hermite normal forms and δ-vector, to appear in J. Comb. Theory

Ser. A, also available at arXiv:1009.6023v1.

[5] T. Hibi, A. Higashitani and Y. Nagazawa, Ehrhart polynomials of convex polytopes with small volume,

European J. Combinatorics 32 (2011), 226–232.

[6] A. Higashitani, Shifted symmetric δ-vectors of convex polytopes, Discrete Math. 310 (2010), 2925–

2934.

[7] R. P. Stanley, “Enumerative Combinatorics, Volume 1,” Wadsworth & Brooks/Cole, Monterey, Calif.,

1986.

[8] R. P. Stanley, Decompositions of rational convex polytopes, Annals of Discrete Math. 6 (1980), 333 –

342.

[9] R. P. Stanley, On the Hilbert function of a graded Cohen–Macaulay domain, J. Pure and Appl. Algebra

73 (1991), 307 – 314.

[10] A. Stapledon, Inequalities and Ehrhart δ-vectors, Trans. Amer. Math. Soc. 361 (2009), 5615–5626.

[11] A. Stapledon, Additive number theorem and inequalities in Ehrhart theory, arXiv:0904.3035v1

[math.CO].

Akihiro Higashitani, Department of Pure and Applied Mathematics, Graduate School of

Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan

E-mail address: a-higashitani@cr.math.sci.osaka-u.ac.jp

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