Gorenstein polytopes with trinomial -polynomials
The characterization of lattice polytopes based upon information about their Ehrhart -polynomials is a difficult open problem. In this paper, we finish the classification of lattice polytopes whose -polynomials satisfy two properties: they are palindromic (so the polytope is Gorenstein) and they consist of precisely three terms. This extends the classification of Gorenstein polytopes of degree two due to Batyrev and Juny. The proof relies on the recent characterization of Batyrev and Hofscheier of empty lattice simplices whose -polynomials have precisely two terms. Putting our theorem in perspective, we give a summary of these and other existing results in this area.
Keywords: Integral polytope, lattice polytope, Ehrhart polynomial, -vector, Gorenstein polytopes, empty simplices.
1.1. Basic notions and terminology
Let us start by setting up notation and recalling the main objects. For an introduction to Ehrhart theory, we refer to .
Let be a lattice polytope of dimension (i.e., is a full-dimensional convex polytope in with vertices in ). Throughout the paper, lattice polytopes are identified if they are isomorphic via an affine lattice-preserving transformation. The Ehrhart series of
is a rational function of the form
where the polynomial appearing in the numerator has nonnegative integer coefficients. We call the polynomial the -polynomial of . The positive integer is the normalized volume of , denoted by ; it is equal to times the usual Euclidean volume of . Note that there are only finitely many lattice polytopes of fixed normalized volume . The highest possible coefficient equals the number of interior lattice points of . The maximal integer such that is called the degree of . We also have the equality . A lattice simplex is an empty simplex if contains no lattice points except for its vertices, equivalently, . Empty simplices appear naturally in singularity theory  and optimization .
There are two well-known higher-dimensional constructions of lattice polytopes. Let denote the convex hull of a subset . For a lattice polytope , we can construct a new lattice polytope
of dimension . This polytope is called the lattice pyramid over . We often use lattice pyramid shortly for a lattice polytope that has been obtained by successively taking lattice pyramids. Note that the -polynomial does not change under lattice pyramids . We also define a lattice polytope to be a Cayley polytope of , if is isomorphic to
where is the standard lattice basis of , see e.g. .
One of the directions in Ehrhart theory is to characterize -polynomials that have an especially simple form and to classify all lattice polytopes with these -polynomials. The motivation is that such results help to understand the restrictions that this important invariant imposes on the structure of a lattice polytope and to learn what to expect in more general situations. In order to put our main theorem in this paper in perspective, we will present some of the existing results in this area.
Small dimensions: Let us describe what is known about -polynomials of small-dimensional lattice polytopes. In dimension , for a given lattice interval of length , we have . In dimension , the -polynomials of lattice polygons have been classified by Scott . It holds that with is the -polynomial of a lattice polygon if and only if
(i.e., has no interior lattice points), or
and (here, is isomorphic to ), or
The upper bound in the last point is often refered to as Scott’s theorem. We refer to  for a thorough discussion.
In dimension there are currently only partial results known. The arguably most significant one is White’s theorem : a three-dimensional lattice simplex is empty (i.e., ) if and only if it is the Cayley polytope of two empty line segments in . Recently, all three-dimensional lattice polytopes with at most lattice points (i.e., ) have been classified [10, 11].
Small degree: It is natural to take the degree of the -polynomial as a measure of complexity. Any degree zero lattice polytope is a unimodular simplex (i.e., the convex hull of affine lattice basis). For degree one, taking lattice pyramids over lattice intervals yields that any -polynomial with arbitrary is possible. Lattice polytopes of degree one are completely classified :
Lattice pyramids over , or
Cayley polytopes of line intervals in .
Lattice polytopes of degree at most two are not yet completely classified. However, their -polynomials are known [24, 14]. A polynomial with is the -polynomial of a lattice polytope (in some dimension) if and only if
(here, is isomorphic to a lattice pyramid over ), or
Note how close this is to the characterization in dimension two above. It follows from the proof in  that any such polynomial can be given by the -polynomial of a lattice polytope in dimension three.
Small number of monomials: An even more general problem is to consider the number of terms in the -polynomial. Batyrev and Hofscheier [6, 7] have recently classified all lattice polytopes whose -polynomials are binomials, i.e., of the form . Let be a -dimensional lattice polytopes with such a binomial -polynomial. Since the degree one case is known, let . Hence, implies that is an empty simplex. It can be observed [7, Prop.1.5] that . Let . In this case, it is proven in  that has -polynomial (with ) if and only if is a Cayley polytope of empty line segments in . Note that for and this recovers White’s theorem. In particular, one sees from [7, Example 2.2] that any and is possible for an -polynomial of the form . The reader might notice the analogy with the degree one case above.
For , we are in an exceptional situation. Let us consider only -polynomials of lattice polytopes that are not lattice pyramids (otherwise, by what we’ve just seen, any can appear). Note that since is a simplex, it follows from  that . Now, the following characterization can be deduced from the results in : (with ) is the -polynomial of a -dimensional lattice polytope with where is not a lattice pyramid if and only if
and is a power of a prime .
It is not hard to see that this implies , in particular, . Hence, there are only finitely many non-lattice-pyramid lattice polytopes with binomial -polynomials for given and arbitrary . They are completely classified by Batyrev and Hofscheier . It turns out that they are uniquely determined by their -polynomial. As their results are the key ingredients in our proof we will describe them in more detail below (see 2.2).
Palindromic -polynomials: A polynomial (with ) is palindromic, if for . A lattice polytope is called Gorenstein, if it has palindromic -polynomial. Equivalently, the semigroup algebra associated to the cone over is a Gorenstein algebra. Gorenstein polytopes are of interest in combinatorial commutative algebra, mirror symmetry, and tropical geometry (we refer to [2, 8, 18]). In each dimension, there exist only finitely many Gorenstein polytopes. Any Gorenstein polytope has a dilate that is a reflexive polytope (in the sense of Batyrev ). They are known up to dimension . For fixed degree, there exist only finitely many Gorenstein polytopes that are not lattice pyramids . They have been completely classified by Batyrev and Juny up to degree two . In particular, their results imply that a polynomial with is the -polynomial of a -dimensional lattice polytope that is not a lattice pyramid if and only if
and , or
and , or
and , or
1.3. Classification of palindromic -trinomials
The main result (Theorem 3.1) of this paper finishes the complete classification of all lattice polytopes that are not lattice pyramids and whose -polynomial is palindromic and has precisely three terms. In the case of degree two, this was already done by Batyrev and Juny . Here, we only consider the case when the degree is strictly larger than two. In this situation, the lattice polytope is necessarily an empty simplex, and we can apply methods and results of Batyrev and Hofscheier [6, 7]. Since the precise formulation of Theorem 3.1 needs some more notation, let us describe here only two immediate consequences. First, the complete characterization of palindromic -trinomials:
Let , and be integers. The polynomial is the -polynomial of a lattice polytope of dimension if and only if the integers satisfy one of the following conditions:
, and ;
, and ;
, and ;
, and , where and ;
, and , where and .
The case was already known, as described in (1) and (2) of the previous section.
Secondly, Theorem 3.1 implies the following uniqueness result:
A lattice simplex that is not a lattice pyramid is uniquely determined by its dimension and its -polynomial if it is of the form with .
Let us note that for any of these lattice simplices that are not lattice pyramids have dimension or , see Theorem 3.1.
1.4. Future work
It is known  that there exists a function in terms of the degree and the leading coefficient of an -polynomial of a lattice polytope such that . In the situation of Corollary 1.1 (where ) one observes that satisfies . In other words,
Moreover, equality implies and so as described in (2) above is isomorphic to a lattice pyramid over . Now, having seen how Scott’s theorem could be generalized from dimension two to degree two , we make the following guess about a more general class of -trinomials:
Let be a lattice polytope with -polynomial and . Then , or equivalently,
1.5. Organization of the paper
In Section 2 we recall the notation and results by Batyrev and Hofscheier. In Section 3 we present and prove the main result of this paper (Theorem 3.1): the classification of Gorenstein polytopes with -trinomials of degree .
We would like to thank Alexander Kasprzyk for discussion. The first author is partially supported by a JSPS Fellowship for Young Scientists and by JSPS Grant-in-Aid for Young Scientists (B) 26800015. The second author is partially supported by the Vetenskapsrådet grant NT:2014-3991.
2. The approach by Batyrev and Hofscheier
2.1. The correspondence to subgroups
For a lattice simplex of dimension with a chosen ordering of the vertices , let
Then is a subgroup of the additive group , here identified with : for and , we define , where for a real number , denotes the fractional part of , i.e., . For a positive integer and , we set . We denote the unit of by and the inverse of by . Note that e.g. . For , we define
It is well known that the coefficients of the -polynomial of the lattice simplex can be computed as follows:
Theorem 2.1 ([7, Theorem 2.3]).
There is a bijection between isomorphism classes of -dimensional lattice simplices with a chosen ordering of their vertices and finite subgroups of . In particular, two lattice simplices , are isomorphic if and only if there exists an ordering of their vertices such that .
We recall the following statement which describes when the lattice simplex is a lattice pyramid in terms of .
Proposition 2.2 ([21, Lemma 2.3]).
Let be a lattice simplex of dimension . Then is a lattice pyramid if and only if there is such that for all .
2.2. The classification of lattice polytopes with binomial -polynomials
First, let us describe their generalization of White’s theorem.
Theorem 2.3 ().
Let and let be a lattice simplex of dimension with which is not a lattice pyramid. Then the following statements are equivalent:
the -polynomial of is ;
is isomorphic to the Cayley polytope of empty simplices of dimension 1;
is cyclic and generated by after reordering, where each is an integer which is coprime to .
Batyrev and Hofscheier use the language of linear codes to consider the case . A linear code over with block length is a subspace of the finite vector space (where is a prime). (an matrix with entries in ) is the generator matrix of such an -dimensional linear code if the rows of form a basis of .
Fix a natural number and a prime number . Let be the number of points in -dimensional projective space over . Consider the -matrix whose columns consist of nonzero vectors from each 1-dimensional subspace of . Then is the generator matrix of the simplex code of dimension over with block length .
Theorem 2.5 ().
Let and let be a lattice simplex of dimension which is not a lattice pyramid. Let the -polynomial of be for some and . Then there exists a prime number such that every non-trivial element of has order . In particular, can be identified with , a linear code over with block length . The order of is equal to , where the positive integer is the dimension of the linear code . The numbers are related by the equation
A generator matrix of the linear code is given (up to permutation of the columns) by the rows in the following -matrix:
where is the generator matrix of the -dimensional simplex code over and (resp. the pair ) is repeated (resp. ) times if (resp. if ).
3. The classification of lattice polytopes with palindromic -trinomials
3.1. The main result
If is a matrix, we denote by the matrix with one additional zero column.
Let and be integers and let be a (necessarily empty) lattice simplex of dimension whose -polynomial is . Assume that is not a lattice pyramid over any lower-dimensional simplex. Then the integers satisfy one of the following:
and or and ;
, and , where and with ;
, and , where and with .
Moreover, in each case, a system of generators of the finite abelian group is the set of row vectors of the matrix which can be written up to permutation of the columns as follows:
where is the generator matrix of the simplex code over of dimension with block length and is the matrix all of whose entries are divided by from those of , and where in above matrix is repeated times.
where is the generator matrix of the simplex code over of dimension with block length and (resp. ) is the matrix all of whose entries are divided by from those of (resp. ), and where in above matrix is repeated times.
In case (b) for and the rows of the following matrix generate of size :
In case (c) for and the rows of the following matrix generate of size :
3.2. Preliminary results
For the proof of Theorem 3.1, we prepare some lemmas. Throughout this section, let be a lattice simplex of dimension whose -polynomial equals with and . Note that is necessarily empty.
For , let . The following equality will be used throughout:
Let be an element whose order is and let be coprime to . Then we have . Hence,
Let , with . By the definition of , we observe that divides , so also . Hence, . Therefore, does not divide , so . ∎
Let be the unique element with . Then,
for any , we have ;
there is no integer and such that .
(a) Since , , and , we have
(b) For any integer and , since by (a), we have . However, by , never happens. ∎
The following proposition is crucial for the proof of Theorem 3.1.
Let be a lattice simplex which is not a lattice pyramid whose -polynomial is with and . Let be the unique element with . Then the order of must be or or or , and up to permutation of coordinates is given as follows:
when its order is ;
when its order is ;
when its order is ;
when its order is .
In particular, the dimension of is if the order of is 2 and otherwise.
Let be the order of . Suppose that or . Then , where is the Eulerian -function. In particular, there exists an integer which is coprime to . By Lemma 3.3 and , we obtain
implying that , a contradiction. Thus, and . Hence, .
: Then each is or 0. From , we have after reordering. Fix and let . Since by Lemma 3.4 (a), we have , where . Hence, . On the other hand, since , we also have . Thus, , i.e., . This means that . Hence, if , then is a lattice pyramid by Proposition 2.2, a contradiction. Thus and we conclude that .
: Then each is or or 0. It follows from and that after reordering. Fix and let . Since , we have , where . Hence, . Similarly, since , we have , where . Hence, . On the other hand, since , we also have . Thus, , i.e., , implying that . Hence we conclude that .
: Then each is or or or 0. For , let . Since , and , we obtain , and , that is,
after reordering. Fix and let . Let for and let . Since , we have the following:
, i.e., ;
, i.e., ;
, i.e., .
In particular, we have . On the other hand, since , we have . Thus we obtain
This means , and thus, . Hence we conclude that after reordering.
: Then each is or 0. For , let . Then
Thus and , that is,
after reordering. Fix and let . Let