aa r X i v : . [ m a t h . C O ] O c t A NOTE ON LATTICE-FACE POLYTOPES AND THEIREHRHART POLYNOMIALS
FU LIU
Abstract.
We give a new definition of lattice-face polytopes by removingan unnecessary restriction in [6], and show that with the new definition, theEhrhart polynomial of a lattice-face polytope still has the property that eachcoefficient is the normalized volume of a projection of the original polytope.Furthermore, we show that the new family of lattice-face polytopes containsall possible combinatorial types of rational polytopes. Introduction A convex polytope is a convex hull of a finite set of points. We often omit “convex”and just write polytope. The face poset of a polytope P is the set of all faces of P ordered by inclusion. We say two polytopes have the same combinatorial type ifthey have the same face poset.The d -dimensional lattice Z d = { x = ( x , . . . , x d ) | x i ∈ Z } is the collection of allpoints with integer coordinates in R d . Any point in a lattice is called a lattice point .For any polytope P and positive integer m ∈ N , we denote by i ( m, P ) the numberof lattice points in mP, where mP = { mx | x ∈ P } is the m th dilated polytope of P. An integral or lattice polytope is a convex polytope whose vertices are all latticepoints. A rational polytope is a convex polytope whose vertices are in Q d . Eug`eneEhrhart [4] discovered that for any d -dimensional rational polytope, i ( P, m ) is aquasi-polynomial of degree d in m, whose period divides the least common multipleof the denominators of the coordinates of the vertices of P. (See [8] for a definition ofquasi-polynomials. We do not include it here because it is irrelevant to this paper.)In particular, if P is an integral polytope, i ( P, m ) is a polynomial. Thus, we call i ( P, m ) the
Ehrhart polynomial of P when P is an integral polytope. See [2, 3] forfurther references to the literature of lattice point counting. Although Ehrhart’stheory was developed in the 1960’s, we still do not have a very good understandingof the coefficients of Ehrhart polynomials for general polytopes except that theleading, second and last coefficients of i ( P, m ) are the normalized volume of P , onehalf of the normalized volume of the boundary of P and 1 , respectively.In [5], the author showed that for any d -dimensional integral cyclic polytope P, we have that(1.1) i ( P, m ) = Vol( mP ) + i ( π ( P ) , m ) = d X k =0 Vol k ( π ( d − k ) ( P )) m k , where π ( k ) : R d → R d − k is the map which ignores the last k coordinates of a point.In [6], the author generalized the family of integral cyclic polytope to a bigger Mathematics Subject Classification.
Primary 05A19; Secondary 52B20.
Key words and phrases.
Ehrhart polynomial, lattice-face, polytope. family of integral polytopes, lattice-face polytopes, and showed that their Ehrhartpolynomials also satisfy (1.1).One question that has been asked often is: how big is the family of lattice-facepolytopes?
The motivation of this paper is to answer this question. We examine thedefinition of lattice-face polytopes given in [6], and notice there is a unnecessaryrestriction. After removing this restriction, we give a new definition of lattice-facepolytopes (Definition 3.1). With this new definition, we have two main results ofthis paper:
Theorem 1.1.
For any lattice-face polytope P , the Ehrhart polynomial of P satis-fies (1.1) . In other words, the coefficient of m k in i ( P, m ) is the normalized volumeof the projection π ( d − k ) ( P ) . Theorem 1.2.
The family of lattice-face polytopes contains all combinatorial typesof rational polytopes. More specifically, given any full-dimensional rational polytope P, there exists an invertible linear transformation φ , such that φ ( P ) is a lattice-facepolytope. We remark that the proof given in [6] for the lattice-face polytopes under the olddefinition would work for proving Theorem 1.1. However, we use a slightly differentapproach in this paper. We will prove Theorem 1.1 by using Proposition 4.2, whichprovides an even bigger family of polytopes whose Ehrhart polynomials satisfy (1.1).See Examples 4.6 and 4.7 for polytopes that are not lattice-face polytopes but arecovered by Proposition 4.2.It follows from Theorem 1.1 and Theorem 1.2 that for any rational polytope P, one can apply a linear transformation φ to P, such that i ( φ ( P ) , m ) is a polynomialhaving the same simple form as (1.1). Thus, all the coefficients of i ( φ ( P ) , m ) havea geometric meaning and are positive. This result provides a possible method toprove positivity conjectures on coefficients of the Ehrhart polynomials of integralpolytopes. Namely, given an integral polytope P, if among all the piecewise lineartransformations φ such that φ ( P ) is a lattice-face polytope, one can find one thatpreserves the lattice, then one can conclude that the coefficients of i ( P, m ) are allpositive. 2.
Preliminaries
We first give some definitions and notation, most of which follows [6], and alsopresent relevant results obtained in [6].All polytopes we will consider are full-dimensional unless otherwise noted, so forany convex polytope P, we denote by d both the dimension of the ambient space R d and the dimension of P. We call a d -dimensional polytope a d -polytope. Wedenote by ∂P the boundary. A d -simplex is a polytope given as the convex hull of d + 1 affinely independent points in R d . For any set S, we denote by conv( S ) the convex hull of all the points in S, andby aff( S ) the affine hull of all the points in S. Recall that the projection π : R d → R d − is the map that forgets the lastcoordinate. For any set S ⊂ R d and any point y ∈ R d − , let ρ ( y, S ) = π − ( y ) ∩ S be the intersection of S with the inverse image of y under π. If S is bounded,let p ( y, S ) and n ( y, S ) be the point in ρ ( y, S ) with the largest and smallest lastcoordinate, respectively. If ρ ( y, S ) is the empty set, i.e., y π ( S ) , then let p ( y, S )and n ( y, S ) be empty sets as well. Clearly, if S is a d -polytope, p ( y, S ) and n ( y, S ) NOTE ON LATTICE-FACE POLYTOPES AND THEIR EHRHART POLYNOMIALS 3 are on the boundary of S. Also, we let ρ + ( y, S ) = ρ ( y, S ) \ n ( y, S ) , and for any T ⊂ R d − , we let ρ + ( T, S ) = S y ∈ T ρ + ( y, S ) . Definition 2.1.
Define
P B ( P ) = S y ∈ π ( P ) p ( y, P ) to be the positive boundary of P ; N B ( P ) = S y ∈ π ( P ) n ( y, P ) to be the negative boundary of P and Ω( P ) = P \ N B ( P ) = ρ + ( π ( P ) , P ) = S y ∈ π ( P ) ρ + ( y, P ) to be the nonnegative part of P. Definition 2.2.
For any facet F of P, if F has an interior point in the positiveboundary of P, then we call F a positive facet of P and define the sign of F to be+1 : sign( F ) = +1 . Similarly, we can define the negative facets of P with associatedsign − . It’s easy to see that F ⊂ P B ( P ) if F is a positive facet and F ⊂ N B ( P ) if F isa negative facet.We write P = F ki =1 P i if P = S ki =1 P i and for any i = j , P i ∩ P j is contained intheir boundaries. If F , F , . . . , F ℓ are all the positive facets of P and F ℓ +1 , . . . , F k are all the negative facets of P, then π ( P ) = ℓ G i =1 π ( F i ) = k G i = ℓ +1 π ( F i ) . Because the usual set union and set minus operation do not count the numberof occurrences of an element, which is important in our paper, from now on we willconsider any polytopes or sets as multisets which allow negative multiplicities.
Inother words, we consider any element of a multiset as a pair ( x , m ) , where m is themultiplicity of element x . Then for any multisets M , M and any integers m, n and i, we define the following operators:a) Scalar product: iM = i · M = { ( x , im ) | ( x , m ) ∈ M } . b) Addition: M ⊕ M = { ( x , m + n ) | ( x , m ) ∈ M , ( x , n ) ∈ M } . c) Subtraction: M ⊖ M = M ⊕ (( − · M ) . Let P be a convex polytope. For any y an interior point of π ( P ) , since π isa continuous open map, the inverse image of y contains an interior point of P. Thus π − ( y ) intersects the boundary of P exactly twice. For any y a boundarypoint of π ( P ) , again because π is an open map, we have that ρ ( y, P ) ⊂ ∂P, so ρ ( y, P ) = π − ( y ) ∩ ∂P is either one point or a line segment. We consider polytopes P which have the property:(2.1) | ρ ( y, P ) | = 1 , ∀ y ∈ ∂π ( P ) , i.e., ρ ( y, P ) always has only one point for a boundary point y. We will see later(Corollary 4.5) that any lattice-face polytope P has the property (2.1). The fol-lowing lemma on polytopes satisfying (2.1) will be used in the proof of Theorem1.1. Lemma 2.3. If P = F ki =1 P i , where all P i ’s satisfy (2.1) , then P satisfies (2.1) aswell. Furthermore, Ω( P ) = L ki =1 Ω( P i ) . Proof.
In Lemma 2.5/(v) of [6], we have already shown that if P and the P i ’s allsatisfy (2.1), then Ω( P ) = L ki =1 Ω( P i ) . Therefore, it is enough to show that thecondition that all P i ’s satisfy (2.1) implies that P satisfies (2.1).For any y ∈ ∂π ( P ) , since P = F ki =1 P i , we have that ρ ( y, P ) = S ki =1 ρ ( y, P i ) . If y ∈ ∂π ( P i ) , then ρ ( y, P i ) has one element. Otherwise if y is not in ∂π ( P i ) , since y FU LIU is a boundary point of π ( P ) , y cannot be a point in the interior of P i , so y P i and ρ ( y, P i ) is the empty set. Hence, in either case, ρ ( y, P i ) has finitely many points,for each i = 1 , . . . , k. Therefore, ρ ( y, P ) has finitely many points and cannot be aline segment. Thus, | ρ ( y, P ) | = 1 . (cid:3) For simplicity, for any set S ∈ R d , we denote by L ( S ) = S ∩ Z d the set of latticepoints in S. A new definition of Lattice-face polytopes
We first recall the old definition of lattice-face polytopes given in [6].
Old definition (Definition 3.1 in [6] ): We define lattice-face polytopes recur-sively. We call a 1-dimensional polytope a lattice-face polytope if it is integral.For d ≥ , we call a d -dimensional polytope P with vertex set V a lattice-face polytope if for any d -subset U ⊂ V, a) π (conv( U )) is a lattice-face polytope, andb) π ( L (aff( U ))) = Z d − . In other words, after dropping the last coordinate ofthe lattice of aff( U ) , we get the ( d − d -polytope with vertex set V, (3.1) any d -subset U of V forms a ( d − d − d −
1) vertices. It isclear that not any rational polytope has this property, e.g., a 4-dimensional cube.Therefore, with the old definition, the family of lattice-face polytopes does notcontain all combinatorial types of rational polytopes. Luckily, we are able to revisethe definition such that the restriction (3.1) does not apply.
Definition 3.1.
We define lattice-face polytopes recursively. We call a 1-dimensionalpolytope a lattice-face polytope if it is integral.For d ≥ , we call a d -dimensional polytope P with vertex set V a lattice-face polytope if for any subset U ⊂ V spanning a ( d − π (conv( U )) is a lattice-face polytope, andb) π ( L (aff( U ))) = Z d − . Remark . We have an alternative definition of lattice-face polytopes, which isequivalent to Definition 3.1. Indeed, a d -polytope on a vertex set V is a lattice-facepolytope if and only if for all k with 0 ≤ k ≤ d − , for any subset U ⊂ V spanninga k -dimensional affine space, π ( d − k ) ( L (aff( U ))) = Z k . To avoid confusion, from now on, we will call lattice-face polytopes defined underDefinition 3.1 in [6] old lattice-face polytopes .Lemma 3.3 in [6] gives properties of an old lattice-face polytope. All but one ofthe properties still hold for lattice-face polytopes under the new definition, and theproofs are similar. We state them here without a proof.
Lemma 3.3.
Let P be a lattice-face d -polytope with vertex set V, then we have: (i) π ( P ) is a lattice-face ( d − -polytope. (ii) mP is a lattice-face d -polytope, for any positive integer m. (iii) π induces a bijection between L ( N B ( P )) (or L ( P B ( P )) ) and L ( π ( P )) . (iv) π ( L ( P )) = L ( π ( P )) . NOTE ON LATTICE-FACE POLYTOPES AND THEIR EHRHART POLYNOMIALS 5 (v)
Let H be a ( d − -dimensional affine space spanned by some subset of V. Then for any lattice point y ∈ Z d − , we have that ρ ( y, H ) is a lattice point. (vi) P is an integral polytope. One might ask what is the relation between the family of old lattice-face poly-topes and the family of newly defined lattice-face polytopes. We have the followinglemma.
Lemma 3.4.
Every old lattice-face polytope is a lattice-face polytope.Proof.
We prove the lemma by induction on d, the dimension of the polytope. If d = 1 , an old lattice-face polytope is a lattice-face polytope by definition. Suppose d ≥ d is a lattice-facepolytope. Let P be an old lattice-face d -polytope with vertex set V. For any subset U ⊂ V spanning a ( d − | U | ≥ d. Forany d -subset W of U, by the definition of old lattice-face polytopes, we have thefollowing: • π (conv( W )) is an old lattice-face polytope. • π ( L (aff( W ))) = Z d − . Using (3.1), W forms a ( d − W ) = aff( U ) ⇒ π ( L (aff( U ))) = Z d − . Let V be the vertex set of π (conv( U )) . For any d -subset U ⊂ V , there ex-ists a d -subset W of U such that π ( W ) = U .
Then we have that π (conv( W )) =conv( π ( W )) = conv( U ) is an old lattice-face polytope. It is easy to check that iffor any d -subset U ⊂ V , we have that conv( U ) is an old lattice-face polytope, thenconv( V ) is an old lattice-face polytope. Therefore, we conclude that π (conv( U )) isan old lattice-face ( d − π (conv( U ))is a lattice-face ( d − P is a lattice-face polytope. (cid:3) It is easy to check that if P is a d -simplex, then P is an old lattice-face polytopeif and only if P is a lattice-face polytope. Therefore, the following propositionfollows from Theorem 3.6 in [6]. Proposition 3.5.
For any P a lattice-face simplex, the number of lattice points inthe nonnegative part of P is equal to the volume of P : |L (Ω( P )) | = Vol( P ) . Proof of Theorem 1.1
We break the proof of Theorem 1.1 into the following two propositions.
Proposition 4.1.
Let P be a lattice-face d -polytope, and P = F ℓi =1 P i be a tri-angulation of P without introducing new vertices. Then each P i is a lattice-face d -simplex. Proposition 4.2.
For any d -polytope P, if P has a triangulation F ℓi =1 P i consistingof lattice-face d -simplices, then Ω( P ) = L ℓi =1 Ω( P i ) . Thus, |L (Ω( P )) | = Vol( P ) . FU LIU
Furthermore, the Ehrhart polynomial of P is given by (1.1) i ( P, m ) = Vol( mP ) + i ( π ( P ) , m ) = d X k =0 Vol k ( π ( d − k ) ( P )) m k . Remark . Note that we allow triangulation involving new vertices other thanvertices of P here. It is implicit in [6] that if P has a triangulation without newvertices consisting of lattice-face simplices, then the Ehrhart polynomial of P satis-fies (1.1). However, since we allow introducing new vertices to form a triangulation,Proposition 4.2 takes care of more cases. See Example 4.7.It is clear that Theorem 1.1 follows from Proposition 4.1 and Proposition 4.2.Proposition 4.1 can be proved directly by checking the definition of lattice-facepolytopes, so we will only give the proof of Proposition 4.2. Before that, we firstprove the following lemma. Lemma 4.4.
Any lattice-face simplex satisfies (2.1) .Proof.
Suppose P is a lattice-face d -simplex that does not satisfy (2.1). There exists y ∈ ∂π ( P ) such that ρ ( y, P ) is a line segment. Let F be a facet of P that contains ρ ( y, P ) and U be the vertex set of F. Then aff( U ) is a ( d − Z d that contains a line L = aff( ρ ( y, P )) . Because π sends L to a point y, the dimension of π (aff( U )) is smaller than d − . Hence, π ( L (aff( U ))) = Z d − . Thiscontradicts part b) in Definition 3.1. (cid:3)
The following corollary follows from Lemma 4.4, Lemma 2.3 and Proposition4.1.
Corollary 4.5.
Any lattice-face polytope satisfies (2.1) .Proof of Proposition 4.2.
By Lemma 2.3 and Lemma 4.4, we immediately haveΩ( P ) = L ℓi =1 Ω( P i ) . Thus, |L (Ω( P )) | = | ℓ M i =1 L (Ω( P i )) | = ℓ X i =1 |L (Ω( P i )) | = ℓ X i =1 Vol( P i ) = Vol( P ) . Hence, using this and Lemma 3.3/(iii), |L ( P ) | = |L (Ω( P )) | + |L ( N B ( P )) | = Vol( P ) + |L ( π ( P )) | . Note that for any positive integer m, the dilation mP has a triangulation F ℓi =1 mP i where each mP i is still a lattice-face d -simplex by Lemma 3.3/(ii). Therefore,(4.1) i ( P, m ) = |L ( mP ) | = Vol( mP ) + |L ( π ( mP )) | = Vol( mP ) + i ( π ( P ) , m ) . Let { F , . . . , F ℓ ′ } be the set of facets of P , . . . , P ℓ that are contained in the negativeboundary N B ( P ) of P. Then we have
N B ( P ) = F ℓ ′ i =1 F i and π ( P ) = π ( N B ( P )) = ℓ G i =1 π ( F i ) . One checks that π ( F i )’s are lattice-face ( d − i ( π ( P ) , m ) in (4.1) with Vol d − ( m π ( P )) + i ( π (2) ( P ) , m ) : i ( P, m ) = Vol( mP ) + i ( π ( P ) , m ) = Vol( mP ) + Vol d − ( m π ( P )) + i ( π (2) ( P ) , m ) . NOTE ON LATTICE-FACE POLYTOPES AND THEIR EHRHART POLYNOMIALS 7
Applying this argument inductively, we obtain i ( P, m ) = d X k =0 Vol k ( m π ( d − k ) ( P )) = d X k =0 Vol k ( π ( d − k ) ( P )) m k . (cid:3) As we mentioned in the introduction, Proposition 4.2 provides a larger family ofpolytopes than the family of lattice-face polytopes which still have Ehrhart poly-nomials satisfying (1.1). We will finish this section with two examples of polytopes P where P is not a lattice-face polytope, but by using Proposition 4.2, we still havethat i ( P, m ) satisfies (1.1).
Example 4.6.
Let P be the polygon with vertices { (0 , , (2 , , (1 , , (3 , } . Onecan check that P is not a lattice-face polytope because π ( L (aff( { (0 , , (3 , } ))) = { z | z ∈ Z } 6 = Z . However, P has a triangulation without introducing new vertices P = conv( { (0 , , (2 , , (1 , } ) ⊔ conv( { (2 , , (1 , , (3 , } ) , where both trianglesare lattice-face simplices. Therefore, by Proposition 4.2, i ( P, m ) = X k =0 Vol k ( π (2 − k ) ( P )) m k = 2 m + 3 m + 1 . Example 4.7.
Let P be the polygon with vertices { v = (0 , , v = (4 , , v =(3 , , v = (1 , } . One can check that P is not a lattice-face polytope because π ( L (aff( { v , v } ))) = { z | z ∈ Z } 6 = Z . There are only two possible triangu-lations of P without introducing new vertices, but neither of them is one con-sisting of lattice-face simplices. However, if we introduce a new vertex v =(2 , , we can obtain the triangulation P = conv( { v , v , v } ) ⊔ conv( { ( v , v , v } ) ⊔ conv( { ( v , v , v } ) ⊔ conv( { ( v , v , v } ) , where all triangles are lattice-face simplices.Therefore, by Proposition 4.2, i ( P, m ) = X k =0 Vol k ( π (2 − k ) ( P )) m k = 15 m + 4 m + 1 . Polytopes in π -general position and the proof of Theorem 1.2 For any linear transformation φ : R d → R d , we can associate a d × d matrix M to φ, such that φ ( x ) = M ( x ) . Therefore, when we describe a linear transformation,we often just describe it by its associated matrix.We denote by diag( A , . . . , A k ) the block diagonal matrix with square matrices A , . . . , A k on the diagonal. In particular, diag( c , . . . , c d ) denotes the d × d diagonalmatrix with diagonal entries c , . . . , c d . Definition 5.1.
We say that a finite set V ⊂ R d is in π -general position if aff( V ) = R d and for any k : 0 ≤ k ≤ d − , and any subset U ⊂ V , we have that(5.1) if aff( U ) is k -dimensional, then π d − k (aff( U )) is k -dimensional.We say that a d -polytope P in π -general position if its vertex set is in π -generalposition . Remark . We can understand property (5.1) in a more algebraic way. In partic-ular, when k = d − , a ( d − H has the property that π ( H ) is ( d − H is nonzero. FU LIU
By the alternative definition of lattice-face polytopes in Remark 3.2, it’s easyto see that a lattice-face polytope is a polytope in π -general position. Therefore,we use rational polytopes in π -general position as the bridge to connect lattice-facepolytopes and general rational polytopes. In fact, the proof of Theorem 1.2 followsfrom the following two propositions. Proposition 5.3.
For any finite set V ⊂ Q d with aff( V ) = R d , there exists aninvertible linear transformation φ associated to an upper triangular matrix withinteger entries and ’s on the diagonal such that φ ( V ) is a finite set of rationalpoints and is in π -general position.In particular, if V is the vertex set of a rational d -polytope, then φ ( P ) is arational polytope in π -general position. Proposition 5.4.
Suppose V ⊂ Q d is a finite set in π -general position. Thenthere exist nonzero integers c , . . . , c d , such that for any U ⊂ V with aff( U ) = R d , we have that φ (conv( U )) is a lattice-face polytope, where φ is the invertible lineartransformation associated with the diagonal matrix diag( c , . . . , c d ) .In particular, if V is the vertex set of P a rational d -polytope in π -general posi-tion, then φ ( P ) is a lattice-face polytope. We prove these propositions in two subsections below. Each proof is precededby a pair of lemmas. It is easy to check that the operator aff commutes with anylinear transformation. We will use this fact often in the proofs.5.1.
Proof of Proposition 5.3.Lemma 5.5.
Suppose V is a finite set in R d such that aff( V ) = R d and for k = d − and any U ⊂ V , (5.1) is satisfied. Then for any k < d − , and any U ⊂ V spanninga k -dimensional affine space, we have that π (aff( U )) is k -dimensional.Proof. Because aff( V ) = R d , there exist v , . . . , v d − − k ∈ V \ U, such that aff( U ∪{ v , . . . , v d − k } ) is ( d − U = U ∪ { v , . . . , v d − − k } . Then we havethat aff( π ( ˜ U )) = π (aff( ˜ U )) has dimension d − . However, the number of elementsin π ( ˜ U ) \ π ( U ) is at most d − − k. Hence, the dimension of π (aff( U )) = aff( π ( U ))is at least k. Since the dimension of aff( U ) is k, we must have that π (aff( U )) is k -dimensional. (cid:3) Lemma 5.6.
Suppose d ≥ and H is a ( d − -dimensional affine space in R d with normal vector n . (i) Suppose v = ( v , . . . , v d ) T is a vector that is not in the null space of n T , i.e., v · n = 0 , and v d = 1 . Let M v be the the d × d upper triangular matrix with ’s on the diagonal, − v i the ( i, d ) -entry for ≤ i ≤ d, and ’s elsewhere.Then π ( M v ( H )) is ( d − -dimensional. (ii) Suppose A is a ( d − × ( d − invertible matrix. Let M A be the d × d block diagonal matrix diag( A, . Then π ( H ) is ( d − -dimensional if andonly if π ( M A ( H )) is ( d − -dimensional.Proof. There exists a ∈ R , such that H = { x | n · x = a } . For any invertiblematrix M, it is easy to check that M ( H ) is the ( d − { x | (( M − ) T n ) · x = a } . Hence, the normal vector of M ( H ) is ( M − ) T n . (i) By Remark 5.2, π ( M v ( H )) is ( d − M v ( H ) is nonzero. However, the normal NOTE ON LATTICE-FACE POLYTOPES AND THEIR EHRHART POLYNOMIALS 9 vector of M v ( H ) is ( M − v ) T n . One can verify that ( M − v ) T is the d × d lower triangular matrix matrix with 1’s on the diagonal, v i the ( d, i )-entryfor 1 ≤ i ≤ d, and 0’s elsewhere. In particular, the last row of ( M − v ) T is v T . Therefore, the last coordinate of ( M − v ) T n is v T n = v · n = 0 . (ii) ( M − A ) T is the block diagonal matrix diag(( A − ) T , . One checks that thelast coordinate of ( M − A ) T · n , the normal vector of M A ( H ), is exactly thelast coordinate of n , the normal vector of H. Therefore, by Remark 5.2, π ( H ) is ( d − π ( M A ( H )) is ( d − (cid:3) Proof of Proposition 5.3.
We prove the proposition by induction on d. When d = 1 , any finite set V ∈ Q d is in π -general position by definition. Assume d ≥ V is a finite set,there are finitely many ( d − V. Suppose they are H , . . . , H ℓ with normal vectors n , . . . , n ℓ , respectively. Foreach i : 1 ≤ i ≤ ℓ, the null space of n Ti is a ( d − R d . Let H the the set of points in R d with the last coordinate equal to 1. The set H = { H i ∩ H = ∅ : 1 ≤ i ≤ ℓ } is a finite set of ( d − H . One sees easily that the complement (with respect to H ) of the unionover H contains lattice points. Therefore, there exists a vector v ∈ Z d ∩ H such that v H i , for each i : 1 ≤ i ≤ ℓ. We pick such a v , then we have that v · n i = 0 for eachi, and the last coordinate of v is 1. Let φ be the invertible linear transformationassociated to M v , where M v is the matrix described in Lemma 5.6/(i). By Lemma5.6/(i), we have that π ( φ ( H i )) is ( d − i : 1 ≤ i ≤ ℓ. Let V = φ ( V ) . Because π ( V ) ⊂ Q d − , by the induction hypothesis, there existsan upper triangular matrix A with integer entries and 1’s on the diagonal, such that A ( π ( V )) is a finite set of rational points and is in π -general position. Let ψ be theinvertible linear transformation associated to the block diagonal matrix diag( A, . It is easy to see that A ◦ π = π ◦ ψ. Let φ = ψ ◦ φ . Because both ψ and φ areupper triangular matrix with integer entries and 1’s on the diagonal, φ is such amatrix too. It is clear that φ ( V ) = ψ ( V ) is a finite set of rational points. We willshow that φ ( V ) = ψ ( V ) is in π -general position.Since both of ψ and φ are invertible, φ is invertible as well. Therefore, aff( φ ( V )) =aff( V ) = R d . We only need to show that for any subset U ⊂ φ ( V ) , (5.1) holds forany k : 0 ≤ k ≤ d − . Suppose aff( U ) is ( d − φ is invertible, we have thataff( φ − ( U )) is ( d − φ − ( U )) = H i , for some i =1 , . . . , ℓ. As discussed above, π ( φ ( H i )) is ( d − π ( φ ( H i )) is ( d − π ( ψ ( φ ( H i )))is ( d − π (aff( U )) = π (aff( φ ( φ − ( U )))) = π ( φ (aff( φ − ( U )))) = π ( φ ( H i )) = π ( ψ ( φ ( H i ))) is ( d − U ) is k -dimensional, where 0 ≤ k < d − . Let U = ψ − ( U ) . Since ψ is invertible, we have that aff( U ) is k -dimensional. Note that U is a subset of V = φ ( V ) . However, by the construction of φ , we know that V is a set sat-isfying the hypothesis in Lemma 5.5. Therefore, by Lemma 5.5, we have thataff( π ( U )) = π (aff( U )) is k -dimensional. Because A is invertible, aff( A ( π ( U ))) is k -dimensional. But A ( π ( U )) is a subset of A ( π ( V )) , which is in π -general posi-tion (in R d − ). Thus, π ( d − − k ) (aff( A ( π ( U )))) is k -dimensional. Therefore, the dimension of π ( d − k ) (aff( U )) = π ( d − − k ) (aff( π ( U ))) = π ( d − − k ) (aff( π ( ψ ( U )))) = π ( d − − k ) (aff( A ( π ( U )))) is k. (cid:3) Proof of Proposition 5.4.Lemma 5.7.
Suppose U ⊂ Q d such that both aff( U ) and π (aff( U )) are ( d − -dimensional. Then there exists a nonzero integer c U such that for any c a nonzeromultiple of c U , we have that π ( L (aff( φ c ( U )))) = Z d − , where φ c is the invertible linear transformation associated with the diagonal matrix diag(1 , . . . , , c ) .Proof. Since U is a set of points in Q d and aff( U ) is ( d − U )can be described by a linear equation: α x + · · · α d x d = a, for some integers α , . . . , α d , a. Because π (aff( U )) is ( d − α d = 0 . Let c U = α d . If c = kc U , for some nonzero k ∈ Z , then aff( φ c ( U ))is the ( d − α x + · · · α d − x d − + k x d = a. For any lattice point y = ( y , . . . , y d − ) ∈ Z d − , the intersection ofaff( φ c ( U )) with the inverse image of y under π is the point ( y , . . . , y d − , k ( a − ( α x + · · · α d − x d − ))) , which is a lattice point. Therefore, π ( L (aff( φ c ( U )))) = Z d − . (cid:3) Lemma 5.8.
Suppose d ≥ . For any finite set V ⊂ R d in π -, the finite set π ( V ) ⊂ R d − is in π -general position.Proof. Because aff( V ) = R d , there exists U ⊂ V such that aff( U ) is ( d − π (aff( U )) is ( d − π ( V )) = π (aff( V )) ⊃ π (aff( U )) has to be ( d − k : 0 ≤ k ≤ d − , let U ′ be a subset of π ( V ) that spans a k -dimensionalaffine space. There exists U ⊂ V such that π ( U ) = U ′ . We need to show that π ( d − − k ) (aff( U ′ )) = π ( d − − k ) (aff( π ( U ))) = π ( d − k ) (aff( U )) is k -dimensional. Since V is in π -general position, it is enough to show that aff( U ) is k -dimensional.Suppose dim(aff( U )) = k. Given that aff( U ′ ) = aff( π ( U )) = π (aff( U )) is k -dimensional, we must have that dim(aff( U )) ≥ k + 1 . There exists W ⊂ U such thatdim(aff( W )) = k + 1 . Note that k + 1 ≤ d − . Because V is in π -general position, π ( d − ( k +1)) (aff( W )) is ( k + 1)-dimensional. This implies that dim( π ( i ) (aff( W ))) = k + 1 , for any i with 0 ≤ i ≤ d − ( k + 1) . In particular, π (aff( W )) = aff( π ( W )) hasdimension k + 1 . However, aff( U ′ ) = aff( π ( U )) ⊃ aff( π ( W )) , but dim(aff( U ′ )) = k < aff( π ( W )) . This is a contradiction. (cid:3)
Proof of Proposition 5.4.
We prove the proposition by induction on d. When d = 1 ,V = { v , . . . , v n } where each v i is a rational number, thus can be written as v i = p i q i for some integers p i and q i = 0 . Let φ = Q ni =1 q i . Clearly, φ ( V ) is a set of integers.For any U ⊂ V with aff( U ) = R , we have that φ (conv( U )) is an integral polytope,thus is a lattice-face polytope.Assume d ≥ V )) is smallerthan d. By Lemma 5.8, π ( V ) ⊂ Q d − is in π -general position in R d − . Thus, thereexists nonzero integers c , . . . , c d − , such that for any U ′ ⊂ π ( V ) with aff( U ′ ) = R d − , we have that conv( ψ ( U ′ )) is a lattice-face polytope, where ψ = diag( c , . . . , c d − ). NOTE ON LATTICE-FACE POLYTOPES AND THEIR EHRHART POLYNOMIALS 11
Let e ψ = diag( c , . . . , c d − ,
1) be the linear transformation in R d corresponding to ψ. Let U ⊂ V such that aff( U ) is ( d − V is in π -generalposition, we have that π (aff( U )) is ( d − ψ and e ψ are invertibleand π ◦ ˜ ψ = ψ ◦ π , aff( e ψ ( U )) = e ψ (aff( U )) and π (aff( e ψ ( U ))) = ψ ( π (aff( U ))) are( d − e ψ ( U ) are still in Q d . Thus, by Lemma 5.7, thereexists a nonzero integer c U such that for any c a nonzero multiple of c U , we havethat π ( L (aff( φ c ( e ψ ( U ))))) = Z d − , where φ c = diag(1 , . . . , , c ). Let c d = Y U ⊂ V : aff( U ) is ( d − c U . We claim the linear transformation φ = diag(1 , . . . , , c d ) ◦ e ψ = diag( c , . . . , c d − , c d )has the desired property. For any U ⊂ V with aff( U ) = R d , we need to check that P U = φ (conv( U )) = conv( φ ( U )) is a lattice-face polytope. It is enough to check thecase when φ ( U ) is the vertex set of P U . We will show this by checking the definitionof lattice-face polytopes. For any subset W ⊂ φ ( U ) spanning a ( d −
1) dimensionalaffine space, we need to show thata) π (conv( W )) is a lattice-face polytope, and b) π ( L (aff( W ))) = Z d − . Since φ is an invertible linear transformation, we have that W := φ − ( W ) ⊂ U ⊂ V and aff( W ) is ( d − V is in π -general position, π (aff( W )) is ( d − π ◦ φ = ψ ◦ π. Thus, π ( W ) = π ( φ ( W )) = ψ ( π ( W )) . Therefore, π (conv( W )) = conv( π ( W )) = conv( ψ ( π ( W ))) . However, since π ( W ) ⊂ π ( V ) , conv( ψ ( π ( W ))) is a lattice-face polytope.b) Since c d is a nonzero multiple of c W , we have that π ( L (aff( W ))) = π ( L (aff( φ ( W )))) = π ( L (aff( φ c d ( e ψ ( W ))))) = Z d − . (cid:3) References
1. T. M. Apostol,
Introduction to analytic number theory , Springer, 1976.2. A. Barvinok,
Lattice points, polyhedra, and complexity , Park City Math Institute Lecture Notes(Summer 2004), to appear.3. M. Beck and S. Robins,
Computing the continuous discretely: Integer-point enumeration inpolyhedra , Springer (to appear), preprint at http://math.sfsu.edu/beck/papers/ccd.html.4. E. Ehrhart,
Sur les poly`edres rationnels homoth´etiques `a n dimensions , C. R. Acad. Sci. Paris (1962), 616–618.5. F. Liu, Ehrhart polynomials of cyclic polytopes , Journal of Combinatorial Theory Ser. A (2005), 111–127.6. ,
Ehrhart polynomials of lattice-face polytopes , Transactions of the AMS (2008),3041–3069.7. I. G. Macdonald,
Polynomials associated with finite cell-complexes , J. London Math. Soc. (1971), 4:181–192.8. R. P. Stanley, Enumerative combinatorics, vol. 1 , Cambridge Studies in Advanced Mathemat-ics, vol. 49, Cambridge University Press, Cambridge, 1997.
Department of Mathematics, One Shields Avenue, University of California, Davis95616.
E-mail address ::