Ehrhart polynomial roots of reflexive polytopes

Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's"canonical line hypothesis". We characterise such polytopes when dim(P)<= 7. We also consider the"half-strip condition", where all roots z satisfy -dim(P)/2<= Re(z)<= dim(P)/2-1, and show that this holds for any reflexive polytope with dim(P)<= 5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi--Shibata in dimension 34.


Introduction
Let P ⊂ Z d ⊗ Z Q be a convex lattice polytope of dimension d. The Ehrhart polynomial [10] L P counts the number of lattice points in successive dilations of P , i.e. L P (m) = mP ∩ Z d for all m ∈ Z ≥0 , and is a polynomial of degree d. Stanley [26] showed that the corresponding generating series, the Ehrhart series, can be written as a rational function with numerator a degree d polynomial whose coefficients define the so-called δ-vector (or h * -vector ) of P : Starting with a δ-vector one can easily recover the Ehrhart polynomial: Lemma 1.1. Let P be a d-dimensional convex lattice polytope with δ-vector (δ 0 , δ 1 , . . . , δ d ). Then In general, combinatorial interpretations for the coefficients δ i of the δ-vector are not known, however the work of Ehrhart [11] tells us that: where P • = P \ ∂P denotes the (strict) interior of P ; (iv) δ 0 + . . . + δ d = d! vol(P ), where vol(P ) denotes the (non-normalised) volume of P .
Hibi's Lower Bound Theorem [15] states that if P • ∩ Z d > 0 then δ 1 ≤ δ i for each 2 ≤ i ≤ d − 1. In particular, combined with (ii) we see that the δ i are positive. A convex lattice polytope P is called reflexive if the dual (or polar ) polyhedron is also a lattice polytope. If P is reflexive then P * is also reflexive, and P • ∩Z d = {0}. Reflexive polytopes are of particular importance in toric geometry: they correspond to Gorenstein toric Fano varieties and are a key combinatorial tool, as introduced by Batyrev [1], for constructing topologically mirror-symmetric pairs of Calabi-Yau varieties. Reflexive polytopes were characterised by Hibi [14] as being those polytopes (up to lattice translation) with palindromic δ-vectors, i.e. δ i = δ d−i for each 0 ≤ i ≤ d. A special class of reflexive polytopes are the smooth Fano polytopes. These are simplicial reflexive polytopes P such that, for each facet F , the vertices vert(F ) of F generate the underlying lattice Z d . They correspond to the smooth toric Fano varieties. For a summary of the various equivalences between lattice polytopes and toric Fano varieties, see [19].
Several recent results have concentrated on the roots of the Ehrhart function L P , regarded as a polynomial over C. These results are inspired in part by Rodriguez-Villegas' study [25] of Hilbert polynomials all of whose roots z ∈ C lie on a line Re(z) = −a/2, and the connection when a = 1 with the Riemann zeta function. Braun [5] has shown that the roots of L P lie inside a disc centred at −1/2 with radius d(d − 1/2). Beck-de Loera-Develin-Pfeifle-Stanley [2] and Braun [6] have also shown that the roots z lie in a strip when d ≤ 5, or for arbitrary d when Im(z) = 0. An example [16] that fails to satisfy the strip condition (S) is known in dimension 15. When P is a reflexive polytope, Macdonald's Reciprocity Theorem [21] gives that L P (−m − 1) = (−1) d L P (m), and so the roots of L P are symmetrically distributed with respect to the line Re(z) = −1/2. In fact: where z 1 , . . . , z d ∈ C are the roots of L P .
Bey-Henk-Wills [3] proved that if (CL) Re(z) = − 1 2 for all roots z of L P then P is a reflexive polytope (after possible translation by a lattice vector). Golyshev [12] proposed the study of Fano varieties satisfying the so-called canonical line hypothesis; in the context of lattice polytopes this is equivalent to the roots satisfying condition (CL). He conjectured that every d-dimensional smooth Fano polytope with d ≤ 5 satisfies the canonical line hypothesis. This was proved in [13], along with an example of a smooth Fano polytope in dimension 6 failing to satisfy (CL).
With the above results in mind, we introduce the following terminology: Definition 1.3. We say that a lattice polytope P is a CL-polytope if the roots z ∈ C of the Ehrhart polynomial L P satisfy (CL). We say that P is real if the imaginary part Im(z) = 0 for all roots z of L P .
Proof. First we consider the case when d = 2k. Let z i ∈ C denote the roots of the Ehrhart polynomial L P . Then .
In § §2-4 we characterise when a reflexive polytope P with d ≤ 5 is a CL-polytope: A characterisation when d = 6 and 7 is given in §5.1, Theorems 5.3 and 5.4. As a consequence we have: We conjecture that Corollary 1.6 holds for all d; if true the result would be sharp, since equality is achieved by the d-dimensional cube {−1, 1} d corresponding to the anticanonical polytope of (P 1 ) d . In § §2-4 we also give characterisations for the real case when d ≤ 5. These characterisations differ from the case of CL-polytopes in Theorem 1.5 simply by flipping both of the (equivalent) inequalities when d = 2 or 3, and by flipping the first inequality when d = 4 or 5. In §5.2, Theorems 5.6 and 5.7, we give characterisations when d = 6 and 7; again these differ from the case for CL-polytopes by flipping certain inequalities. It was conjectured in [22] that for any reflexive polytope P , the roots z of L P satisfy the half-strip condition In §7 we prove the following: Theorem 1.7. Let P be a d-dimensional reflexive polytope and let z ∈ C be a root of L P . Inequality (HS) holds when d ≤ 5, or for arbitrary d when P is real.
This result halves the bounds of (S) in the case of reflexive polytopes. Ohsugi-Shibata [23] have found a 34-dimensional reflexive polytope failing to satisfy (HS). In §7.2 we give a general method for determining whether a palindromic δ-vector with δ 1 = 1 arises from a lattice polytope, and use this in §7.3 to give a 10-dimensional example that fails to satisfy (HS). This corresponds to a terminal Gorenstein fake weighted projective space. It seems probable that Theorem 1.7 also holds when d = 6 and 7, and possible that d = 10 is the smallest dimension in which (HS) fails, although we do not prove this.
A hierarchy of hypotheses. Golyshev [12] introduced two additional bounds on the roots z ∈ C of the Ehrhart polynomial L P , which he called the canonical strip hypothesis and the narrowed canonical strip hypothesis. These correspond, respectively, to: By a slight modification of the proof of Proposition 1.4 above, we obtain: Proposition 1.8. Let P be a d-dimensional real reflexive polytope such that the roots z ∈ C of L P satisfy (CS). Then vol(P ) ≥ 2 d .
We have a hierarchy of implications These hypotheses provide meaningful ways of partitioning the space of reflexive polytopes (or, more generally, Fano polytopes) by their δ-vectors. In higher dimensions, where the number of reflexive polytopes is vast, this becomes an essential tool for studying their classification.
Proof. We only give a proof of the last assertion. It follows from Lemma 1.1 that The roots of L P are Since we have that δ 1 ≥ 6 in case (ii), the roots of L P (z) satisfy (CS).
Proof. It follows from Lemma 1.1 that On the other hand, P is a CL-polytope if and only if there exists b ∈ R such that By comparing the constant term in the two expressions, we see that P is a CL-polytope if and only if there exists b ∈ R such that 4(δ 1 + 1)b 2 + δ 1 − 23 = 0.
Consequently, P is a CL-polytope if and only if δ 1 ≤ 23. A reflexive polytope P is real if and only if there exists a ∈ R such that Again, comparing the constant term gives (1 + δ 1 )a 2 + (1 + δ 1 )a + 6 = 0, hence P is real if and only if δ 1 ≥ 23. Moreover, since the solutions to this quadratic are given by we see that −1 < a < 0 and (CS) is satisfied.
Example 3.3. Case (i) in Theorem 3.2 can certainly occur: the 4-dimensional cube {−1, 1} 4 is a reflexive polytope with δ-vector (1, 76, 230, 76, 1). Notice that this is not the only polytope with this δ-vector: a second example is given in Example 3.5. In general it would be an interesting problem to classify all polytopes P with δ-vector equal to the d-dimensional cube {−1, 1} d , that is, with Ehrhart polynomial L P (m) = (2m + 1) d .

Theorem 3.2 tells us that
In particular we have that vol(P ) ≤ 2 4 and P ∩ Z 4 ≤ 3 4 .
Example 3.4. We shall calculate the δ-vector for the d-dimensional cube {−1, 1} d . In general let P and Q be lattice polytopes such that L P (m) = L Q (2m) for all m ∈ Z ≥0 , and let (δ 0 , δ 1 , . . . , δ d ) be the δ-vector of Q. Then hence the δ-vector of P is given by It is well-known that the δ-vector of Q can be expressed in terms of the Eulerian numbers, corresponding to the anticanonical divisor −K X has: Equivalently, the value of h 0 (X P , −K X ) is equal to the coefficient of t 12 in the Taylor expansion We have that δ 1 = 76 and δ 2 = 230, and P is both a CL-polytope and a real polytope.
The second and third inequalities in Corollary 3.6(ii) also appear in the work of Bey-Henk-Wills [3, Proposition 1.9(ii)], however they overlook the first inequality. As noted above, this inequality is necessary in order to specify which of the regions we are interested in, although the precise form this inequality takes is a matter of choice. In Theorem 3.7 we will show that flipping this inequality corresponds to selecting the regions containing the real reflexive polytopes.
. Consider the direct product P = Q × Q. This is a 4-dimensional polytope with Ehrhart polynomial Hence P has δ-vector (1, 95, 294, 95, 1) and this gives equality in the third expression in Theorem 3.7(ii), i.e. (95, 294) is an integer point on the parabola defined by the discriminant ∆.
Example 3.9. We investigate which δ-vectors lie on the parabola 17( Since we already know an integer point, δ 1 = 76, δ 2 = 230, by the "slope method" we can easily parameterise the rational points on the curve: Interpreting the first of these equations as a quadratic in γ, and restricting to δ 1 ∈ Z, we see that δ 1 + 5 is a square; this is equivalent to saying that P ∩ Z 4 is a square. Setting δ 1 + 5 = P ∩ Z 4 = N 2 for some N ∈ Z ≥1 and solving for γ, we obtain: The second equation gives that δ 2 − 5 is a square, and setting δ 2 − 5 = M 2 for some M ∈ Z ≥1 we obtain: Equating these two expressions for γ, and remembering that P ∩ Z 4 ≥ 6, we find that By consulting the Kreuzer-Skarke classification [20] we see that the cases with δ 2 = (2N + 3) 2 + 5 never occur (this corresponds to the upper branch of the parabola); the reflexive polytopes lie on the bottom branch with δ 2 = (2N − 3) 2 + 5, for each N ∈ {3, . . . , 13}. When N > 13 there are no matching δ-vectors.
The occurring δ-vectors are recorded in Table 1.
When the first condition in (3.3) is satisfied, and recalling that δ 1 ≤ δ 2 , we see that Hence we obtain δ 2 > 230, so we are done. When the second condition in By the proof of Proposition 3.12, P is a 4-dimensional real reflexive polytope with vol(P ) = 3 if and only if its δ-vector equals (1, 1, 68, 1, 1). However, by the method described in §7.2 below, we can show that no such reflexive polytope exists.
The 4-dimensional reflexive polytopes were classified by Kreuzer-Skarke [20]: there are 473 800 776 cases. Corollary 3.10 makes extracting the real reflexive polytopes a simple matter, and we can recover their δ-vectors. We find that the region to the left of the parabola (and closest to the δ 2 -axis) in Figure 1 is empty: all the δ-vectors lie in the narrow region between the parabola and the tangent. A plot of all of the δ-vectors suggests very strongly that there is an additional inequality awaiting discovery. Furthermore: Proposition 3.13. Let P be a 4-dimensional real reflexive polytope. Then the roots of L P satisfy (CS).
As a consequence, Proposition 1.8 tells us that vol(P ) ≥ 2 4 . Unfortunately we do not have a theoretical explanation for Proposition 3.13.

Remaining cases.
Proposition 3.15. Let P be a 4-dimensional reflexive polytope, and suppose that there are two roots of L P which are real, and that there are two roots with real part −1/2 (i.e. we are in case (c) above). Then vol(P ) ≥ 4/3.

Dimension five
Let P be a 5-dimensional reflexive polytope with δ-vector (1, δ 1 , δ 2 , δ 2 , δ 1 , 1). Then one of the roots of L P is −1/2. The remaining roots of L P fall into the four possible cases (a)-(d) described at the beginning of §3.
Since −1/2 is a root, we know that L P is divisible by z + 1/2. Set f (z) := 5!L P (z)/(z + 1/2). Substituting z = −1/2 + βi in f and multiplying through by 5! gives This is a quadratic in β 2 , and G(β) = 0 if and only if L P (−1/2 + βi) = 0. The two cases follow from Lemma 3.1. By combining the first two inequalities in case (ii) we see that δ 1 < 237 and δ 2 < 1682; including case (i) we have that δ 1 ≤ 237 and δ 2 ≤ 1682. Once more we see that the inequalities in Theorem 4.1(ii) are determined by the discriminant ∆ of G: the parabola 41(δ 1 + 9δ 2 − 9) 2 = 2(41δ 1 + 96) 2 + 2(41δ 2 − 85) 2 . The tangent at the point (δ 1 , δ 2 ) = (237, 1682) is given by 71δ 1 = 9δ 2 + 1689, and together these two equations (or, more accurately, the corresponding two inequalities) cut out three regions in the positive quadrant. The inequality δ 2 < 7δ 1 +23 distinguishes which of these regions contains the δ-vectors for the CL-polygons and, as we shall see in Theorem 4.4 below, which contains the real reflexive polygons. The situation is essentially the same as the case in 4-dimensions illustrated in Figure 1. As in the 4-dimensional case, we expect there to be a "missing inequality" that excludes the top-left-most region from consideration.
By the proof of Proposition 4.6, P is a 5-dimensional real reflexive polytope with vol(P ) = 16/5 if and only if its δ-vector equals (1, 1, 190, 190, 1, 1). One can show that no such reflexive polytopes exist by the method of §7.2 below.

Remaining cases.
Proposition 4.7. Let P be a 5-dimensional polytope, and suppose that there are two roots of L P which are real, and that there are two roots with real part −1/2. Then vol(P ) ≥ 19/20.

(4.3)
When the first condition of (4.3) is satisfied, we have: < 81 We conclude that r + r cos θ as required.

Dimensions six and seven
If P is a (2k + 1)-dimensional reflexive polytope then −1/2 is a root of L P . This we shall consider both 6-and 7-dimensional reflexive polytopes together. We shall require the following lemma; this is simply an application of Descartes' rule of signs. Proof. First suppose that all roots of (5.1) are non-negative real numbers. Then it is easy to see that one of the conditions (i), (ii), or (iii) is satisfied. Conversely, suppose that one of (i), (ii), or (iii) holds. Let B ′ > 0, C ′ < 0, and D ′ > 0 denote a small perturbation of, respectively, the coefficients B ≥ 0, C ≤ 0, and D ≥ 0. Then it follows from Rouche's Theorem that the roots of are very close to the roots of (5.1), and Lemma 5.1 implies that all roots of (5.2) are positive. This perturbation method give us a sequence of roots of a polynomial sequence, and the limit of this sequence of roots is the roots of (5.1).
In § §5.1-5.2 we will make use of the following two polynomials:
Proof. Again, the proof parallels that of Theorem 5.4. In this case we obtain the cubic in α 2 given by g 7 (α 2 ), where Since ∆(f 7 ) = ∆(g 7 ), the result follows by Lemma 5.2.
We have already shown that these conjectures hold when d ≤ 7. Slightly less confidently, we suggest: Conjecture 6.3. Let P be a d-dimensional real reflexive polytope. Then vol(P ) ≥ 2 d .
In §3.2 we shown that this conjecture holds when d = 4, however this is achieved by using the classification of Kreuzer-Skarke [20] to show that the roots of L P satisfy (CS) for every 4-dimensional real reflexive polytope P , and lacks a theoretical explanation. Experimentation suggests the following: Conjecture 6.4. Let P be a 6-or 7-dimensional reflexive polytope, and suppose that there exists a root −1/2 + a + bi ∈ C of L P , where a = 0 and b = 0. Then |a| < 5/2.
If this conjecture holds then we can conclude that hypothesis (HS) holds when d ≤ 7; however see §7 for an example when d = 10 whose roots fails to satisfy (HS).
7. Hypothesis (HS) and an example in dimension ten 7.1. Hypothesis (HS). In this section we prove Theorem 1.7. We actually prove a stronger result: Theorem 7.1. Let P be a d-dimensional reflexive polytope and let α ∈ C be a root of L P .
We have already shown that case (ii) holds when d = 2 and d = 3 (Propositions 2.1 and 2.2, respectively). When d = 4 the roots of L P (z) fall into the four cases (a)-(d) described at the beginning of §3. Of these, (a)-(c) satisfy (ii): either a root α is of the form −1/2 ± bi, and so Re(α) = −1/2, or α is real and so is covered by (i). The only remaining possibility is (d), in which case Proposition 3.16 gives the result. Similarly, when d = 5 the result follows from Proposition 4.8.

7.2.
Realising a simplex from a δ-vector. In §7.3 we give a 10-dimensional reflexive polytope failing to satisfy (HS). Finding an integer-valued palindromic vector whose numerics give an example is straight-forward; the difficulty lies in showing that this vector is the δ-vector for a lattice polytope. The method we describe below can be used to find all lattice polytopes P with palindromic vector of the form (1, 1, δ 2 , . . . , δ ⌊d/2⌋ , . . . , δ 2 , 1, 1) of length d + 1, if such P exist, and otherwise to prove that there are no such P .
Let N ′ ⊂ Z d be the sublattice generated by vert(P ). Then G = Z d /N ′ and the order of G is given by the index mult(P ) := Z d : N ′ , which we call the multiplicity of P . If we restrict P to the sublattice N ′ , we recover (up to isomorphism) the simplex Q associated with the weighted projective space Y [4, Proposition 2]. Moreover, there exists a matrix H in Hermite normal form with det(H) = mult(P ) such that Hence vol(P ) = mult(P ) · vol(Q). In particular, Thus the possible choices of weights for Q are greatly restricted, and can easily be listed for any given h using (7.4) and (7.5). Suppose now that we have chosen some possible weights (λ 0 , . . . , λ d ) for Q. These weights must satisfy three conditions, as follows.
(i) The inequalities given by (7.5 where {a/b} denotes the fractional part of a/b ∈ Q. (ii) Since P is reflexive, we have that d! vol(P * ) ∈ Z. But d! vol(P * ) = d! vol(Q * )/ mult(P ), and d! vol(Q * ) is simply the anticanonical degree of Y , given by Hence we have the requirement that: where L P is determined by the target δ-vector via Lemma 1.1. The δ-vector for Q can be easily computed [18]: Hence the Ehrhart polynomial L Q can be computed and the condition verified. Notice that the leading coefficient of L Q is, by construction, at most equal to the leading coefficient of L P (since this are equal to vol(Q) and vol(P ) respectively), hence (7.6) is automatically satisfied for all sufficiently large values of m.
In practice, these three conditions are sufficiently strong as to exclude many candidate choices of weights, and this is often sufficient to show that no such Q exists, or to restrict the possibilities to only one or two choices of weights. Finally, for each choice of weights (λ 0 , . . . , λ d ) satisfying the above conditions, one can simply work through the possible Hermite normal forms H with det(H) = mult(P ) and consider the resulting simplicies arising from (7.2). Here one can exploit the symmetries of Q arising from the weights, and the fact that Q is reflexive, in order to reduce the number of choices of H that need to be considered.
Finally, we need to compute the resulting polytope P and check that the δ-vector agrees. This is trivial. We find that, up to isomorphism, where e i is the i-th standard basis element, and that P has δ-vector (1, 1, 1, 1, 9, 28, 9, 1, 1, 1, 1).