Yusuke SUYAMA

1 Introduction

Let P⊂Rdbe an integral convex polytope of dimension d,that is,a convex polytope whose vertices have integer coordinates.For a non-negative integer l,we write lP={lx|x∈P}.Ehrhart[2]proved that the number of lattice points in lP can be expressed by a polynomial in l of degree d:|(lP)∩ Zd|=cdld+cd−1ld−1+···+c0.This polynomial is called the Ehrhart polynomial of P.It is known that:

(1)c0=1.

(2)cd−1is half of the sum of relative volumes of facets of P(see[1,Theorem 5.6]).

(3)cdis the volume of P(see[1,Corollary 3.20]).

However,we have no formula on other coefficients of Ehrhart polynomials.In particular,we do not know a formula on c1for a general 3-dimensional integral convex polytope.In this paper,we find an explicit formula on c1of the Ehrhart polynomial of a 3-dimensional simple integral convex polytope(see Theorem 2.1).

Pommersheim[4]gave a method for computing the(d−2)-nd coefficient of the Ehrhart polynomial of a d-dimensional simple integral convex polytope P by using toric geometry.He obtained an explicit description of the Ehrhart polynomial of a tetrahedron by using this method.Our formula is obtained by using this method for a general 3-dimensional simple integral convex polytope.

The structure of the paper is as follows.In Section 2,we state the main theorem and give a few examples.In Section 3,we give a proof of the main theorem.

2 The Main Theorem

Let P ⊂ R3be a 3-dimensional simple integral convex polytope,and let F1,···,Fnbe the facets of P.For k=1,···,n,we denote by vk∈ Z3the inward-pointing primitive normal vector of Fk.For an edge E of P,we denote by Vol(E)the relative volume of E,that is,the length of E measured with respect to the lattice of rank one in the line containing E.

definition 2.1For each edgeE=Fk1∩Fk2ofP,we define an integerm(E)and a rational numbers(E)as follows:

(1)We definem(E)=|((Rvk1+Rvk2)∩Z3)/(Zvk1+Zvk2)|.

(2)There exists a basise1,e2for(Rvk1+Rvk2)∩Z3such thatvk1=e1andvk2=pe1+qe2for someq＞p≥0.Then we defines(E)=s(p,q),wheres(p,q)is the Dedekind sum,which is defined by

Remark 2.1 We have q=m(E).Although p is not uniquely determined,s(p,q)does not depend on the choice of e1,e2.Thus s(E)is well-defined.

definition 2.2For each facetFofP,we define a rational numberC(F)as follows.We name vertices and facets aroundFas in Figure1.We denote byv∈Z3the inward-pointing primitive normal vector ofF.

Figure 1 Vertices and facets around F.

Fori=1,···,r,we define

wherevk0=vkr,vkr+1=vk1, ε0= εr,P0=Pr,Pr+1=P1,Q0=Qrand〈·,·〉is the standard inner product onR3.Then we define

wherePj−1Pjis the edge whose endpoints arePj−1andPj,and the determinants above are understood to be one whenj=i+1.

Remark 2.2 The proof of Theorem 2.1 below shows that C(F)does not depend on the choice of Fk1.

The following is our main theorem.

Theorem 2.1LetP⊂R3be a3-dimensional simple integral convex polytope,and letE1,···,EmandF1,···,Fnbe the edges and the facets ofP,respectively.Then the coefficientc1of the Ehrhart polynomial|(lP)∩Z3|=c3l3+c2l2+c1l+c0is given by

Example 2.1 Let a,b,c be positive integers with gcd(a,b,c)=1 and let P⊂R3be the tetrahedron with vertices

We put A=gcd(b,c),B=gcd(a,c),C=gcd(a,b)and d=ABC.Then we have the following table:

Table 1 The values of Vol(E),s(E)and C(F)

Thus we have which coincides with the formula in[4,Theorem 5].

Example 2.2 Let a and c be positive integers and b be a non-negative integer.Consider the convex hull P⊂R3of the six points

P is a 3-dimensional simple polytope.We put g=gcd(b,c).Then we have the following table:

Table 2 The values of Vol(E),s(E)and C(F)

Figure 2 The simple polytope P.

Thus we have

On the other hand,since

for z=0,1,···,cl,we have

The coefficient of l is also

3 Proof of Theorem 2.1

First we recall some facts about toric geometry(see[3]for details).Let P⊂Rdbe a d-dimensional integral convex polytope.We define a cone

for each face F of P.Then the set

of such cones forms a fan in Rd,which is called the normal fan of P.Let X(ΔP)be the associated projective toric variety.We denote by V(σ)the subvariety of X(ΔP)corresponding to σ ∈ ΔP.Let Tdi(X(ΔP)) ∈ Ai(X(ΔP))Qbe the i-th Todd class in the Chow group of i-cycles with rational coefficients.

Theorem 3.1LetP⊂Rdbe ad-dimensional integral convex polytope and|(lP)∩Zd|=cdld+cd−1ld−1+ ···+c0be its Ehrhart polynomial.IfTdi(X(ΔP))has an expression of thewhere[V(σF)]is the class ofV(σF)in the Chow group andVol(F)is the relative volume ofF.

Now we assume that d=3 and P is simple.Then the associated toric variety X(ΔP)is Q-factorial and we know the ring structure of the Chow ring A∗(X(ΔP))Qwith rational coefficients.Let E1,···,Emand F1,···,Fnbe the edges and the facets of P,respectively.We have

If Fk1and Fk2are distinct,then

in A∗(X(ΔP))Q.

Pommersheim gave an expression of Tdd−2(X(ΔP))for a d-dimensional simple integral convex polytope P⊂Rd.In the case where d=3,we have the following theorem.

Theorem 3.2(see[4])IfP⊂R3is a3-dimensional simple integral convex polytope,then

We use the notation in definition 2.2.It suffices to show

for 2 ＜ s≤ t＜ r and D(s,t)=1 for s＞ t.define u ∈ (Q3)∗by 〈u,v〉=1,〈u,vk1〉=0, 〈u,vk2〉=0.By(3.1)and(3.2),we have

Hence it suffices to show

for any j=3,···,r.

First we claim that

for any j=2,···,r − 1.By Cramer’s rule,we have

So we have

Taking the inner product of both sides of(3.5)withgives

which meansdet(vkj+1,vkj,vkj−1).Taking the inner product of both sides of(3.5)withgives

which meansdet(v,vkj+1,vkj−1).Thus(3.4)follows.

We show(3.3)by induction on j.If j=3,then both sides are a2ε2.If j=4,then both sides are a2b3ε2ε3+a3ε3.Suppose 4 ≤ j ≤ r−1.By(3.4)and the hypothesis of induction,we have

On the other hand,

Since

we have

Thus(3.3)holds for j+1.This completes the proof of Theorem 2.1.

AcknowledgementsThe author wishes to thank his supervisor,Prof.Mikiya Masuda,for his continuing support.

[1]Beck,M.and Robins,S.,Computing the Continuous Discretely,Undergraduate Texts in Mathematics,Springer-Verlag,New York,2007.

[2]Ehrhart,E.,Polynˆomes Arithm´etiques et M´ethode des Poly`edres en Combinatoire,Birkh¨auser,Boston,Basel,Stuttgart,1977.

[3]Fulton,W.,Introduction to Toric Varieties,Annals of Mathematics Studies,131,Princeton Univ.Press,Princeton,NJ,1993.

[4]Pommersheim,J.E.,Toric varieties,lattice points and Dedekind sums,Math.Ann.,295,1993,1–24.