Universal inequalities in Ehrhart Theory
aa r X i v : . [ m a t h . C O ] M a r UNIVERSAL INEQUALITIES IN EHRHART THEORY
GABRIELE BALLETTI AND AKIHIRO HIGASHITANI
Abstract.
In this paper, we show the existence of universal inequalities for the h ∗ -vector of a latticepolytope P , that is, we show that there are relations among the coefficients of the h ∗ -polynomial whichare independent of both the dimension and the degree of P . More precisely, we prove that the coefficients h ∗ and h ∗ of the h ∗ -vector ( h ∗ , h ∗ , . . . , h ∗ d ) of a lattice polytope of any degree satisfy Scott’s inequalityif h ∗ = 0. Introduction
Background and main result.
Let N ∼ = Z d be a lattice of rank d and let N R := N ⊗ Z R ∼ = R d . Wesay that a convex polytope P ⊂ N R is a lattice polytope if its vertices are all elements of N . Two latticepolytopes P, Q ⊂ N R are said to be unimodularly equivalent if there exists an affine lattice automorphism ϕ ∈ GL d ( Z ) ⋉ Z d of N such that ϕ R ( P ) = Q . In what follows, unless stated otherwise, we consider latticepolytopes as being defined up to unimodular equivalence.Given a lattice polytope P of dimension d , one can associate an enumerative function k
7→ | kP ∩ N | ,counting the number of lattice points in the k -th dilation kP of P , where k is a positive integer. Ehrhart [4]proved that this function is interpolated by a polynomial, i.e. that there exists a polynomial ehr P , calledthe Ehrhart polynomial of P , such that ehr P ( k ) = | kP ∩ N | for any k ≥
1. Moreover, its generatingfunction is known to be the rational function X k ≥ ehr P ( k ) t k = P i ≥ h ∗ i t i (1 − t ) d +1 , where h ∗ i = 0 for any i ≥ d +1. We call the sequence of integers ( h ∗ , h ∗ , . . . , h ∗ d ) appearing in the numeratorof the generating function the h ∗ -vector (or δ -vector ) of P , and the polynomial h ∗ P ( t ) = h ∗ + h ∗ t + · · · + h ∗ s t s the h ∗ -polynomial (or δ -polynomial ) of P , where s is the degree of this polynomial. We define s to be the degree of P . In the following, we will sometimes use the notation h ∗ i = h ∗ i ( P ) when we want to specifythe polytope P . The constant term h ∗ of the h ∗ -polynomial is always 1. Moreover, the coefficients areall nonnegative integers (see [9]). The h ∗ -polynomial (the h ∗ -vector) is an important and meaningfulinvariant for P , and despite (in fixed dimension) it is equivalent to the Ehrhart polynomial of P , it isoften preferred to ehr P as its coefficients have a well-understood combinatorial interpretation. For moredetails, see Section 2 (Proposition 2.1).The following question is one of the most important unsolved problems in Ehrhart Theory. Question 1.1.
Can one characterize the polynomials with nonnegative integer coefficients that are the h ∗ -polynomial of some lattice polytope? This question has a trivial answer in dimension one, while an answer in dimension two has been givenby Scott [8].
Theorem 1.2 (Scott [8] (1976)) . A polynomial h ∗ ( t ) = 1 + h ∗ t + h ∗ t ∈ Z ≥ [ t ] is the h ∗ -polynomial ofa lattice polytope of dimension two if and only if it satisfies one of the following conditions:(i) h ∗ = 0 ;(ii) h ∗ ≤ h ∗ ≤ h ∗ + 3 ;(iii) h ∗ = 7 and h ∗ = 1 .Moreover, the case (iii) is satisfied only by the h ∗ -polynomial of the triangle conv( { , e , e } ) , where e and e are a basis for the ambient lattice and is its origin. Mathematics Subject Classification.
Primary: 52B20; Secondary: 52B12.
Key words and phrases.
Lattice polytope, Ehrhart polynomial, h ∗ -polynomial, universal inequalities, spanningpolytopes. All the cases for d ≥ Theorem 1.3 ([14, Theorem 2]) . Let P be a lattice polytope of degree at most two. Then its h ∗ -polynomial h ∗ P ( t ) = 1 + h ∗ t + h ∗ t satisfies one of the following conditions:(i) h ∗ = 0 ;(ii) h ∗ ≤ h ∗ + 3 ;(iii) h ∗ = 7 and h ∗ = 1 .Moreover, the case (iii) is satisfied only by the h ∗ -polynomial of lattice simplices obtained via multiplelattice pyramid constructions (see (3) ) over the triangle conv( { , e , e } ) , where e and e are part ofa basis for the ambient lattice and is its origin. Henk–Tagami [5, Proposition 1.10] proved that those are sufficient conditions, i.e. that any polynomialof degree two in Z ≥ [ t ] satisfying one of the conditions (i)–(iii) of Theorem 1.3 is the h ∗ -polynomial ofsome lattice polytope of degree two. Note that the inequality h ∗ ≤ h ∗ of Theorem 1.2, coming from (1),does not appear in Theorem 1.3, as the dimension of the polytope may be greater than two.We call the inequalities (i)–(iii) in Theorem 1.3 Scott’s inequality . Our main result is the followingfurther generalization of Theorem 1.3 to polynomials h ∗ ( t ) = 1 + h ∗ t + h ∗ t + · · · ∈ Z ≥ [ t ] of any degreesatisfying h ∗ = 0. Theorem 1.4 (Main Theorem) . Let P be a lattice polytope whose h ∗ -polynomial h ∗ P ( t ) = 1 + h ∗ t + h ∗ t + · · · ∈ Z ≥ [ t ] satisfies h ∗ = 0 . Then h ∗ P ( t ) satisfies Scott’s inequality, i.e. it satisfies one of the followingconditions:(i) h ∗ = 0 ;(ii) h ∗ ≤ h ∗ + 3 ;(iii) h ∗ = 7 and h ∗ = 1 .Moreover, the case (iii) is satisfied only by the h ∗ -polynomial of lattice simplices obtained via multiplelattice pyramid constructions over the triangle conv( { , e , e } ) and then considered as lattice polytopeswith respect to a refined lattice. The latter part of the statement of Theorem 1.4 will be clarified and proven in Proposition 5.1.Theorem 1.4 gives the first relation among the coefficients of the h ∗ -polynomial of a lattice polytope P which is valid independently of both the dimension and the degree of P . We call this new kind ofinequality universal . The existence of this kind of inequalities has been conjectured by Benjamin Nill(private communication), who also suggested the term “universal”.With the following example, we notice that the condition h ∗ = 0 in Theorem 1.4 is necessary. Example 1.5.
The 5-dimensional lattice polytopeconv( { , e , e , e + e + 2 e , e , e + 9 e } ) ⊂ R , where e , . . . , e are a basis for Z , has 1 + 8 t + t + 8 t as its h ∗ -polynomial. In particular it does notsatisfy Scott’s inequality.1.2. Organization of the paper.
The paper is organized as follows. Section 2 is devoted to givingsome backgrounds on Ehrhart Theory and known inequalities for h ∗ -vectors of lattice polytopes. Theproof of Theorem 1.4 will be given in the remaining sections. In Section 3, we prove the main statementof the main theorem for all cases excluding a special one. Such special case needs a technical proof givenin Section 4. Finally, in Section 5, the last part of the statement of Theorem 1.4 will be proven. Acknowledgments
The authors would like to thank the PhD advisor of the first author, Benjamin Nill, for posing thequestion of the existence of universal inequalities, for posing Question 2.2, and, together with JohannesHofscheier, for many inspiring discussions. The project started during a visit of the first author atOtto-von-Guericke Universit¨at, Magdeburg. The actual collaboration started while the authors wereparticipants in the “Einstein Workshop on Lattice Polytopes” in Berlin. Moreover, the authors would
NIVERSAL INEQUALITIES IN EHRHART THEORY 3 like to thank Takayuki Hibi and Akiyoshi Tsuchiya in Osaka University for their hospitality. The firstauthor is partially supported by the Vetenskapsr˚adet grant NT:2014-3991 and the second author ispartially supported by JSPS Grant-in-Aid for Young Scientists (B) ♯ Inequalities and universal inequalities in Ehrhart Theory
Known inequalities in Ehrhart Theory.
The following proposition summarizes some of thewell-known interpretations of the h ∗ -vectors of a lattice polytope. All the following can be deduced fromEhrhart’s original approach [4]. Proposition 2.1.
Let P be a d -dimensional lattice polytope of degree s with respect to the lattice N .Then the h ∗ -vector ( h ∗ , . . . , h ∗ d ) of P satisfies h ∗ = 1 , h ∗ = | P ∩ N | − d − , h ∗ s = | ( d + 1 − s ) P ◦ ∩ N | , h ∗ d = | P ◦ ∩ N | , and d X i =0 h ∗ i = Vol( P ) , where by P ◦ we denote the relative interior of P and by Vol( P ) we denote the normalized volume of P which equals d ! times the Euclidean volume of P . In particular, from the descriptions of h ∗ and h ∗ d , onegets h ∗ d ≤ h ∗ . (1)In the last decades several other relations among the h ∗ -vectors have been proven. Let P be a d -dimensional lattice polytope of degree s with its h ∗ -vector h ∗ ( P ) = ( h ∗ , . . . , h ∗ d ). In [6] Hibi provedthat h ∗ d − + · · · + h ∗ d − i ≤ h ∗ + · · · + h ∗ i +1 for i = 1 , . . . , d − h ∗ + · · · + h ∗ i ≤ h ∗ s + · · · + h ∗ s − i for i = 0 , , . . . , s. Another result from Hibi [6], which is valid in the case P has interior points (which is equivalent to h ∗ d > h ∗ ≤ h ∗ i for i = 1 , , . . . , d − . More recently Stapledon [12, 13] showed the existence of infinite new classes of inequalities and improvedexisting ones.All the mentioned and previously known families of inequalities for the h ∗ -vectors of lattice polytopeshave different forms when specialized to different dimensions or degrees. Namely, they are not universal.2.2. Realizing h ∗ -polynomial. The most natural way to approach Question 1.1 is to fix a polynomial h ∗ ( t ) ∈ Z ≥ [ t ] and try to check whether it is realizable as the h ∗ -polynomial of a lattice polytope. As afirst step one can try to fix a polynomial h ( t ) = 1 + h ∗ t + · · · + h ∗ k t k ∈ Z ≥ [ t ] for some k > h ∗ -polynomial of a lattice polytope by adding parts of degreestrictly larger than k . It is then natural to ask the following question, which firstly has been posed byBenjamin Nill (private communication). Question 2.2.
Let h ( t ) = 1+ h ∗ t + · · · + h ∗ k t k ∈ Z ≥ [ t ] for some k > . Is there a polynomial H ( t ) ∈ Z ≥ [ t ] such that h ( t ) + t k +1 H ( t ) is the h ∗ -polynomial of a lattice polytope? None of the known inequalities for the h ∗ -vectors forbids such a possibility. Furthermore, in Proposi-tion 2.5, we answer positively to Question 2.2 for k ≤ h ∗ -vectors as just described is the following construction. Lemma 2.3 ([5, Lemma 1.3]) . Let P ⊂ R m and Q ⊂ R n be two lattice polytopes. Then the join P ⋆ Q := conv { ( x , n , , ( m , y ,
1) : x ∈ P, y ∈ Q } ⊂ R n + m +1 , where i denotes the origin in R i , has the h ∗ -polynomial h ∗ P ⋆Q ( t ) = h ∗ P ( t ) h ∗ Q ( t ) . Moreover we are going to need a special class of lattice simplices having binomial h ∗ -polynomial ofarbitrary degree. Such family was described by Batyrev–Hofscheier [1, Theorem 2.5]. Lemma 2.4.
For each choice of positive integers s ≥ and b ≥ , there exists a (2 s − -dimensionallattice simplex ∆ s,b having the h ∗ -polynomial h ∗ ∆ s,b ( t ) = 1 + bt s . G. BALLETTI AND A. HIGASHITANI
Proposition 2.5.
Let h ( t ) = 1 + at + bt ∈ Z ≥ [ t ] . Then there exists a polynomial H ( t ) ∈ Z ≥ [ t ] suchthat h ( t ) + t H ( t ) is the h ∗ -polynomial of a lattice polytope. In particular, Question 2.2 is true for k ≤ .Proof. We can choose H ( t ) := ab . Indeed, by Lemma 2.4 there exists a three-dimensional lattice simplex∆ ,b having the h ∗ -polynomial h ∗ ∆ ,b ( t ) = 1 + bt . Let L a be the lattice segment (1-dimensional latticepolytope) L a := conv( { , a + 1 } ) ⊂ R . Then the join ∆ ,b ⋆ L a has the h ∗ -polynomial h ∗ ∆ ,b ⋆L a ( t ) =(1 + bt )(1 + at ) = 1 + at + bt + abt by Lemma 2.3, as required. (cid:3) On the other hand Theorem 1.4 answers negatively to all the other cases.
Corollary 2.6.
Question 2.2 is false for k ≥ . Exploiting the construction used in Proposition 2.5, one can easily create infinite families of polytopesof arbitrarily large degree satisfying Scott’s inequality of Theorem 1.4.
Proposition 2.7.
For each choice of s ≥ and nonnegative integers h ∗ , h ∗ satisfying the conditions (i)–(iii) of Theorem 1.4, there exists a lattice polytope of degree s for which Theorem 1.4 agrees.Proof. By [5, Proposition 1.10], there exists a polytope Q of degree at most two having the h ∗ -polynomial h ∗ Q ( t ) = 1+ h ∗ t + h ∗ t . By Lemma 2.3 and Lemma 2.4, and for any choice of k ≥
1, the join ∆ s − ,k ⋆Q hasthe h ∗ -polynomial h ∗ ∆ s − ,k ⋆Q ( t ) = (1+ kt s − )(1+ h ∗ t + h ∗ t ) = 1+ h ∗ t + h ∗ t + kt s − + kh ∗ t s − + kh ∗ t s . (cid:3) One might be tempted to suspect that the only interesting cases for which Theorem 1.4 applies,are these polytopes artificially built via joins, as in the previous proposition. This is not the case: inExample 5.2 we build a lattice polytope of degree five satisfying Scott’s inequality that cannot be obtainedvia a join construction. 3.
The non-Lawrence prism case
In this section we recall the key notion of our proof from [7] and give a proof of Theorem 1.4 for themajority of the cases, namely for all the polytopes whose spanning polytope (see definition below) is nota Lawrence prism (see (3.2)). The Lawrence prism case needs technical proofs and will be treated inSection 4.3.1.
Spanning polytopes.
Let P ⊂ R d be a d -dimensional lattice polytope with respect to the lattice N of rank d . Let us denote by e N the affine sublattice of N generated by the points of P ∩ N , i.e. e N consists of all the integral affine combinations of P ∩ N . We define the spanning polytope e P associated to P as the lattice polytope given by the vertices of P with respect to the lattice e N and we say that P is spanning if e N = N .We denote by h ∗ e P ( t ) = 1 + f h ∗ t + · · · + f h ∗ ˜ s t ˜ s the h ∗ -polynomial of e P . Then we have the followinginequalities f h ∗ = h ∗ and f h ∗ i ≤ h ∗ i for i ≥ , (2)and in particular ˜ s ≤ s . (See [7, Section 3.2].)We use the following recent result by Hofscheier–Katth¨an–Nill. Theorem 3.1 ([7, Theorem 1.3]) . The h ∗ -polynomial h ∗ ˜ P ( t ) = 1 + f h ∗ t + · · · + f h ∗ ˜ s t ˜ s of any spanningpolytope e P satisfies f h ∗ i ≥ for all i = 0 , . . . , ˜ s. In particular the spanning polytope e P of a lattice polytope P having h ∗ = 0 has degree at most two.3.2. Characterization of lattice polytopes with degree one.
We recall the work by Batyrev–Nill[2]. Given a d -dimensional lattice polytope Q ⊂ R d with respect to Z d , we define the lattice pyramid Pyr( Q ) as the ( d + 1)-dimensional polytopePyr( Q ) := conv( Q × { } ∪ { (0 , . . . , , } ) ⊂ R d +1 . (3)The lattice pyramid construction preserves the h ∗ -polynomial, i.e. h ∗ Q ( t ) = h ∗ Pyr( Q ) ( t ) ([3, Theorem 2.4]).We say that a d -dimensional lattice polytope P is an exceptional simplex if P can be obtained via the NIVERSAL INEQUALITIES IN EHRHART THEORY 5 ( d − P ∼ = Pyr( · · · (Pyr(conv( { , e , e } ))) · · · ) . We say that a d -dimensional lattice polytope P is a Lawrence prism with heights a , . . . , a d − if thereexist nonnegative integers a , . . . , a d − such that P ∼ = conv( { , a e d , e , e + a e d , . . . , e d − , e d − + a d − e d } ) . Theorem 3.2 ([2, Theorem 2.5]) . Let P be a lattice polytope. Then deg( P ) ≤ if and only if P is anexceptional simplex or a Lawrence prism. This classification is a powerful tool. Indeed, as we will see in Proposition 3.3, the case in which thespanning polytope e P of P has degree one is the only one that cannot be entirely deduced directly fromTheorem 3.1 and Theorem 1.3. For this special case the classification result is necessary.3.3. A proof for lattice polytopes whose spanning polytope is not a Lawrence prism.Proposition 3.3.
Let P be a lattice polytope whose h ∗ -polynomial h ∗ P ( t ) = 1 + h ∗ t + h ∗ t + · · · satisfies h ∗ = 0 . Assume that the spanning polytope e P of P is not a Lawrence prism. Then the inequalities (i) – (iii) in Theorem 1.4 hold.Proof. Since f h ∗ ≤ h ∗ by (2) and h ∗ = 0, we have f h ∗ = 0. Then Theorem 3.1 implies that f h ∗ i = 0 for any i ≥
3, in particular e P has degree ˜ s ≤ s = 2, by Theorem 1.3, we see that the h ∗ -vector of e P satisfies Scott’s inequality. By f h ∗ = h ∗ and0 < f h ∗ ≤ h ∗ , we conclude that P also satisfies Scott’s inequality. If ˜ s = 1, then, by our assumptions andTheorem 3.2, e P must be an exceptional simplex. In particular we get h ∗ = f h ∗ = 3, so Scott’s inequalityis satisfied for any value of h ∗ . Finally, if ˜ s = 0 then f h ∗ = 0, so Scott’s inequality is satisfied for any valueof h ∗ , as required. (cid:3) The Lawrence prism case
In this section, we consider the missing case in which P is a d -dimensional lattice polytope with h ∗ ( P ) = 0 and whose spanning polytope e P is a Lawrence prism. To prove this case we first show that,if ∆ ′ and ∆ ′′ are two full-dimensional empty simplices contained in P , then h ∗ (∆ ′ ) = h ∗ (∆ ′′ ). Thisfollows from Proposition 4.1. Above, we call a simplex S empty if it has no lattice points other than itsvertices, or equivalently, if it satisfies h ∗ ( S ) = 0. Later, with an inclusion-exclusion argument, we show(Proposition 4.2) strong conditions on the coefficients h ∗ ( P ) and h ∗ ( P ), which are enough to finish theproof of the main statement of Theorem 1.4.For the proof, we recall some notation from the following well-known technique for the computation ofthe h ∗ -vectors of lattice simplices by associating them to finite abelian groups. Let ∆ be a lattice simplexwith respect to the lattice N and let v , . . . , v d ∈ N be the vertices of ∆. We defineΛ ∆ := ( ( r , . . . , r d ) ∈ [0 , d +1 : d X i =0 r i v i ∈ N, d X i =0 r i ∈ Z ≥ ) . We see that Λ ∆ is a finite abelian group with its addition α + β = ( { α + β } , . . . , { α d + β d } ) ∈ [0 , d +1 for α, β ∈ Λ ∆ , where { r } = r − ⌊ r ⌋ denotes the fractional part of r ∈ R . Note that = (0 , . . . , ∈ Λ ∆ .We define Λ ( h )∆ := ( ( r , . . . , r d ) ∈ Λ ∆ : d X i =0 r i = h ) for h = 0 , , . . . , d. (4)Note that Λ ∆ = F di =0 Λ ( i )∆ and the h ∗ -vector of ∆ can be computed by h ∗ i = | Λ ( i )∆ | for each i, (5)see [3, Corollary 3.11].As in Section 3, we denote by N the ambient lattice of P , while e N ⊆ N is the lattice affinely spannedby the points in P ∩ N . Let e , . . . , e d be a basis for a lattice e N . Since e P is a Lawrence prism, we assumethat P ⊂ D , where D is the unbounded prism D := conv( { , e , . . . , e d − } ) × R e d ⊂ R d . G. BALLETTI AND A. HIGASHITANI
Note that the ( d − { , e , . . . , e d − } ) over which the D is built may not beunimodular simplex if considered with respect to N . Proposition 4.1.
With the notation just introduced, let ∆ ′ and ∆ ′′ be two d -dimensional empty simplicescontained in D , with vertices on e N , but considered as lattice polytopes with respect to the refined lattice N . Suppose that h ∗ (∆ ′ ) = h ∗ (∆ ′′ ) = 0 . Then h ∗ (∆ ′ ) = h ∗ (∆ ′′ ) .Proof. Note that any empty simplex ∆ in D having vertices on e N can be easily described. In particular,there exist 0 ≤ k ≤ d and nonnegative integers c , . . . , c d − such that ∆ can be written as∆ = ∆ ′ k := conv( { e + c e d , e + c e d , . . . , e d − + c d − e d , e k + ( c k − e d } ) , where by e we denote the origin of R d . In the following, all the polytopes are considered with respectto the refined lattice N . We first prove that(a) h ∗ (∆ ′ k ) = h ∗ (∆ k ), where ∆ k := conv( { e , . . . , e d − , e k + e d } );and we conclude by proving(b) h ∗ (∆ k ) = h ∗ (∆ )for any k . The simplices ∆ ′ k , ∆ k and ∆ are represented in Figure 1. e k e = ∆ ′ k e k e = ∆ k e = ∆ Figure 1.
The empty simplices ∆ ′ k , ∆ k and ∆ in the unbounded prism D .Let Λ ∆ k = ( ( r , . . . , r d ) ∈ [0 , d +1 : d − X i =0 r i e i + r d ( e k + e d ) ∈ N, d X i =0 r i ∈ Z ) , Λ ∆ ′ k = (cid:26) ( r , . . . , r d ) ∈ [0 , d +1 : X ≤ i ≤ d − i = k r i ( e i + c i e d ) + r k ( e k + ( c k − e d ) + r d ( e k + c k e d ) ∈ N, d X i =0 r i ∈ Z (cid:27) . Let Λ ( i )∆ k and Λ ( i )∆ ′ k be as in (4) for i = 0 , , . . . , d .We first show that there exists a bijection π between Λ ∆ k and Λ ∆ ′ k mapping Λ (2)∆ k to Λ (2)∆ ′ k . Let( r , . . . , r d ) be any element of Λ ∆ k , we define π : Λ ∆ k → Λ ∆ ′ k by setting π (( r , . . . , r d )) =( π ( r ) , . . . , π ( r d )):= r , . . . , r k − , ( d − X i =0 r i c i + r d ( c k − ) , r k +1 , . . . , r d − , ( r k − r d ( c k − − d − X i =0 r i c i )! . NIVERSAL INEQUALITIES IN EHRHART THEORY 7
We now check that π is well-defined. It is straightforward to verify that P di =0 π ( r i ) becomes an integerwhen P di =0 r i is an integer. Let ( r , . . . , r d ) ∈ Λ ∆ k . By definition, P d − i =0 r i e i + r d ( e k + e d ) ∈ N . Thus, X ≤ i ≤ d − i = k π ( r i )( e i + c i e d ) + π ( r k )( e k + ( c k − e d ) + π ( r d )( e k + c k e d ) ≡ X ≤ i ≤ d − i = k r i ( e i + c i e d ) + d − X i =0 r i c i + r d ( c k − ! ( e k + ( c k − e d ) + r k − r d ( c k − − d − X i =0 r i c i ! ( e k + c k e d )= X ≤ i ≤ d − i = k r i e i + r k e k + r d ( e k + e d ) ∈ N mod e N. Similarly, we can also construct the inverse map π − : Λ ∆ ′ k → Λ ∆ k . We set π − (( r , . . . , r d )) =( π − ( r ) , . . . , π − ( r d )):= r , . . . , r k − , ( r k + r d (1 − c k ) − d − X i =0 r i c i ) , r k +1 , . . . , r d − , ( − r k + d − X i =0 r i c i + r d c k )! . Also in this case the map is well-defined. Note that 0 ≤ π ( r k ) , π ( r d ) , π − ( r ′ k ) , π − ( r ′ d ) <
1. This meansthat, for h ≥ π (Λ ( h )∆ k ) ⊂ Λ ( h − ′ k ∪ Λ ( h )∆ ′ k ∪ Λ ( h +1)∆ ′ k and π − (Λ ( h )∆ ′ k ) ⊂ Λ ( h − k ∪ Λ ( h )∆ k ∪ Λ ( h +1)∆ k . In particular, since by our assumptions Λ (1)∆ k = Λ (3)∆ k = Λ (1)∆ ′ k = Λ (3)∆ ′ k = ∅ , we see that π induces a bijectionbetween Λ (2)∆ k and Λ (2)∆ ′ k . This proves (a).We prove (b) similarly. Let ψ : Λ ∆ → Λ ∆ k be the map defined by ψ (( r , . . . , r d )) := ( { r + r d } , r , . . . , r k − , { r k − r d } , r k +1 , . . . , r d ) . Then P di =0 ψ ( r i ) ∈ Z if P di =0 r i ∈ Z . Let ( r , . . . , r d ) ∈ Λ ∆ . Then P di =0 r i e i ∈ N . Moreover, d − X i =0 ψ ( r i ) e i + ψ ( r d )( e k + e d ) ≡ d − X i =1 r i e i − r d e k + r d ( e k + e d ) = d X i =1 r i e i ∈ N mod e N. Hence, ψ : Λ ∆ → Λ ∆ k is well-defined. We can also construct the inverse ψ − : Λ ∆ k → Λ ∆ by ψ − (( r ′ , . . . , r ′ d )) := (cid:0) { r ′ − r ′ d } , r ′ , . . . , r ′ k − , { r ′ k + r ′ d } , r ′ k +1 , . . . , r ′ d (cid:1) . Exactly as in case (a), although ψ might not induce a bijection between Λ ( h )∆ and Λ ( h )∆ k for h >
3, abijection is induced for h = 2. This proves (b). (cid:3) We can now prove that Theorem 1.4 holds also in the Lawrence prism case.
Proposition 4.2.
Let P be a d -dimensional lattice polytope whose h ∗ -polynomial is of the form h ∗ P ( t ) =1 + h ∗ t + h ∗ t + · · · with h ∗ = 0 . Suppose that its spanning polytope e P is a Lawrence prism. Then h ∗ isdivisible by h ∗ + 1 . In particular, the inequalities (i)–(iii) in Theorem 1.4 hold.Proof. Since e P is a Lawrence prism, e P is of the form e P = conv( { e i , e i + a i e d : 0 ≤ i ≤ d − } ) ⊂ R d , where a , a , . . . , a d − are nonnegative integers and e denotes the origin of R d . We regard e P as a latticepolytope with respect to the sublattice e N ⊆ N , affinely spanned by the points of P ∩ N . We know thatthe h ∗ -polynomial h ∗ e P of e P has degree at most one, i.e. h ∗ e P = 1 + f h ∗ t . Note that since f h ∗ = h ∗ the degreezero case is trivial, so we can assume h ∗ > a , a , . . . , a d − . We define the lattice simplex ∆ i,j for 0 ≤ i ≤ d − ≤ j ≤ a i as the lattice polytope∆ i,j := conv( { e , . . . , e i − , e i + j e d , e i + ( j − e d , e i +1 + a i +1 e d , . . . , e d − + a d − e d } ) , considered with respect to the lattice N . Then ∆ i,j ⊂ P for any i, j . Hence, by monotonicity [11,Theorem 3.3], we have h ∗ (∆ i,j ) = 0. Moreover, ∆ i,j is an empty simplex, equivalently, h ∗ (∆ i,j ) = 0.Therefore, it follows from Proposition 4.1 that all h ∗ (∆ i,j )’s are equal for any i, j . Let c := h ∗ (∆ i,j ). G. BALLETTI AND A. HIGASHITANI
Let ℓ = min { i : a i > } for fixed a , a , . . . , a d − . Let P ′ be the lattice polytope P ′ := conv( { e i , e i + a ′ i e d : 0 ≤ i ≤ d − } ) ⊂ P, considered with respect to N , where a ′ i := a i for i = ℓ and a ′ ℓ := a ℓ −
1. Equivalently, P ′ is the latticepolytope given as (the closure of) P \ ∆ ℓ,a ℓ . For simplicity we introduce the notation ∆ := ∆ ℓ,a ℓ . Let S be the ( d − P ′ ∩ ∆. Then, from the inclusion-exclusionformula ehr P ( k ) = ehr P ′ ( k ) + ehr ∆ ( k ) − ehr S ( k ), it follows that d X i =0 h ∗ i t i = d X i =0 h ∗ i ( P ′ ) t i + d X i =0 h ∗ i (∆) t i − (1 − t ) d − X i =0 h ∗ i ( S ) t i . In particular we obtain the following: h ∗ = h ∗ ( P ′ ) + h ∗ (∆) − h ∗ ( S ) + 1 ,h ∗ = h ∗ ( P ′ ) + h ∗ (∆) − h ∗ ( S ) + h ∗ ( S ) ,h ∗ = h ∗ ( P ′ ) + h ∗ (∆) − h ∗ ( S ) + h ∗ ( S ) . By h ∗ = 0 and by monotonicity, we obtain that h ∗ ( P ′ ) = h ∗ (∆) = h ∗ ( S ) = h ∗ ( S ) = 0. Moreover, wealso know h ∗ (∆) = h ∗ ( S ) = 0. In addition, h ∗ (∆) = c by definition. Therefore, h ∗ = h ∗ ( P ′ ) + 1 and h ∗ = h ∗ (∆) + c. Now, we iterate this construction. More precisely, one can replace P by P ′ and perform the samecomputation for this new starting polytope. We can do this until P ′ becomes ∆ d ′ , , where d ′ = max { i : a i > } . This process stops after b − b = P d − i =0 a i . Hence, we eventually obtain thefollowing: h ∗ = b − h ∗ = bc. This says that h ∗ is divisible by h ∗ + 1, as desired. (cid:3) Necessary condition for (iii)Finally, we study necessary conditions for condition (iii) of Theorem 1.4.
Proposition 5.1.
Let P be a d -dimensional lattice polytope with h ∗ -polynomial t + t + h ∗ t + · · · .Then its spanning polytope e P is unimodular equivalent to the polytope obtained by the ( d − -fold iterationsof the lattice pyramid construction over the triangle conv( { , e , e } ) .Proof. By Theorem 3.1, the h ∗ -polynomial of e P is of the form h ∗ e P = 1 + 7 t + f h ∗ t . Suppose that f h ∗ = 0,then since e P cannot be an exceptional simplex, it must be a Lawrence prism. Then h ∗ is divisible by h ∗ + 1 by Proposition 4.2. In particular, ( h ∗ , h ∗ ) = (7 ,
1) never happens. Since 0 < f h ∗ ≤ h ∗ = 1, weassume that f h ∗ = 1. By Theorem 1.3, e P is unimodularly equivalent to a polytope obtained via multiplelattice pyramids over conv( { , e ′ , e ′ } ), where e ′ , e ′ are two elements of a basis for the affine sublattice e N . (cid:3) This completes the proof of Theorem 1.4.We conclude by noting that polytopes satisfying condition (iii) of Theorem 1.4 are not necessarilyobtained via a join.
Example 5.2.
Let v = , v = 3 e , v = 3 e and v i = e i for i = 3 , . . . ,
8, where e i is a basisfor Z , and let ∆ = conv( { v i : i = 0 , , . . . , } ) be a lattice simplex with respect to the lattice N = Z · (1 / , / , / , / , / , / , / , /
2) + Z . ThenΛ ∆ = h (1 / , / , , , . . . , , (1 / , , / , , . . . , i ⊕ h (0 , / , / , . . . , / i , where h α, . . . , β i denotes the abelian group generated by α, . . . , β . Thus, h ∗ ∆ ( t ) = 1 + 7 t + t + 6 t + 3 t by (5). Notice that ∆ cannot be a join of some two lower-dimensional lattice polytopes by Lemma 2.3. NIVERSAL INEQUALITIES IN EHRHART THEORY 9
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Department of Mathematics, Stockholm University, SE-
106 91
Stockholm, Sweden
E-mail address : [email protected] (A. Higashitani) Department of Mathematics, Kyoto Sangyo University, 603-8555, Kyoto, Japan
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