Interior Lattice Points Research Articles

A new invariant of lattice polytopes

The maximal degree of monomials belonging to the unique minimal system of monomial generators of the canonical module $\omega (K[\mathcal{P}])$ of the toric ring $K[\mathcal{P}]$ defined by a lattice polytope $\mathcal{P}$ will be studied. It is shown that if $\mathcal{P}$ possesses an interior lattice point, then the maximal degree is at most $\dim \mathcal{P} - 1$, and that this bound is the best possible in general.

Castelnuovo Polytopes

It is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound on the genus of a projective curve. Polarized varieties whose sectional genus achieve this bound are called Castelnuovo. On the other hand, a lattice polytope is called Castelnuovo if the associated polarized toric variety is Castelnuovo. Kawaguchi characterized Castelnuovo polytopes having interior lattice points in terms of their h∗-vectors. In this paper, as a generalization of this result, we present a characterization of all Castelnuovo polytopes. Finally, as an application of our characterization, we give a sufficient criterion for a lattice polytope to be IDP.

Deep Lattice Points in Zonotopes, Lonely Runners, and Lonely Rabbits

Abstract Let $P \subseteq {\mathbb {R}}^{d}$ be a polytope and let $\textbf {w}$ be an interior point of $P$. The coefficient of asymmetry$\operatorname {ca}(P,\textbf {w}):= \min \{ \lambda \geq 1: \textbf {w} - P \subseteq \lambda (P - \textbf {w}) \}$ of $P$ about $\textbf {w}$ has been studied extensively in the realm of Hensley’s conjecture on the maximal volume of a $d$-dimensional lattice polytope that contains a fixed positive number of interior lattice points. We zero in on the coefficient of asymmetry for lattice zonotopes, that is, Minkowski sums of line segments with integer endpoints. Our main result gives the existence of an interior lattice point for which the coefficient of asymmetry is bounded above by an explicit constant in $\Theta (d \log \log d)$, for any lattice zonotope that has an interior lattice point. Our work is both inspired by and feeds on Wills’ lonely runner conjecture from Diophantine approximation: we make intensive use of a discrete version of this conjecture (which, in fact, has been proved), and reciprocally, we reformulate the lonely runner conjecture in terms of the coefficient of asymmetry for certain lattice zonotopes.

On the maximum dual volume of a canonical Fano polytope

Abstract We give an upper bound on the volume $\operatorname {vol}(P^*)$ of a polytope $P^*$ dual to a d-dimensional lattice polytope P with exactly one interior lattice point in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp and achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional Fano toric variety X with at worst canonical singularities.

Moduli dimensions of lattice polygons

Given a lattice polygon P with g interior lattice points, we can associate to \(P\) two moduli spaces: the moduli space of algebraic curves that are non-degenerate with respect to \(P\) and the moduli space of tropical curves of genus g with Newton polygon P. We completely classify the possible dimensions such a moduli space can have in the tropical case. For non-hyperelliptic polygons, the dimension must be between g and \(2g+1\) and can take on any integer value in this range, with exceptions only in the cases of genus 3, 4, and 7. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from g to \(2g-1\). In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to P.

Lattice Size and Generalized Basis Reduction in Dimension Three

The lattice size of a lattice polytope P was defined and studied by Schicho, and Castryck and Cools. They provided an “onion skins” algorithm for computing the lattice size of a lattice polygon P in $$\mathbb R^2$$ based on passing successively to the convex hull of the interior lattice points of P. We explain the connection of the lattice size to the successive minima of $$K=(P+(-P))^*$$ and to the lattice reduction with respect to the general norm that corresponds to K. It follows that the generalized Gauss algorithm of Kaib and Schnorr (which is faster than the “onion skins” algorithm) computes the lattice size of any convex body in $$\mathbb R^2$$ . We extend the work of Kaib and Schnorr to dimension three, providing a fast algorithm for lattice reduction with respect to the general norm defined by a convex origin-symmetric body $$K\subset \mathbb R^3$$ . We also explain how to recover the successive minima of K and the lattice size of P from the obtained reduced basis and therefore provide a fast algorithm for computing the lattice size of any convex body $$P\subset \mathbb R^3$$ .

The moduli space of Harnack curves in toric surfaces

Abstract In 2006, Kenyon and Okounkov Kenyon and Okounkov [12] computed the moduli space of Harnack curves of degree d in ${\mathbb {C}\mathbb {P}}^2$ . We generalise their construction to any projective toric surface and show that the moduli space ${\mathcal {H}_\Delta }$ of Harnack curves with Newton polygon $\Delta $ is diffeomorphic to ${\mathbb {R}}^{m-3}\times {\mathbb {R}}_{\geq 0}^{n+g-m}$ , where $\Delta $ has m edges, g interior lattice points and n boundary lattice points. This solves a conjecture of Crétois and Lang. The main result uses abstract tropical curves to construct a compactification of this moduli space where additional points correspond to collections of curves that can be patchworked together to produce a curve in ${\mathcal {H}_\Delta }$ . This compactification has a natural stratification with the same poset as the secondary polytope of $\Delta $ .

Newton polytopes and algebraic hypergeometric series

Let $X$ be the family of hypersurfaces in the odd-dimensional torus ${\mathbb T}^{2n+1}$ defined by a Laurent polynomial $f$ with fixed exponents and variable coefficients. We show that if $n\Delta$, the dilation of the Newton polytope $\Delta$ of $f$ by the factor $n$, contains no interior lattice points, then the Picard-Fuchs equation of $W_{2n}H^{2n}_{\rm DR}(X)$ has a full set of algebraic solutions (where $W_\bullet$ denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.

Sectional genus and the volume of a lattice polytope

For a convex lattice polytope having at least one interior lattice point, a lower bound for its volume is derived from Hibi’s lower bound theorem for the $$h^{*}$$ -vector. On the other hand, it is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound for the genus of a projective curve. In this paper, we prove the equivalence of these two bounds. Namely, a polarized toric variety has maximal sectional genus if and only if its associated polytope has minimal volume. This is a generalization of the known fact that polytopes corresponding to the anticanonical bundles of Gorenstein toric Fano varieties are reflexive polytopes (whose typical examples are minimal volume polytopes with only one interior lattice point).

Tropical hyperelliptic curves in the plane

Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice points collinear. We prove that hyperelliptic graphs can only arise from such polygons. Along the way, we will prove certain graphs do not embed tropically in the plane due to entirely combinatorial obstructions, regardless of whether their metric is actually hyperelliptic.

Enumeration of Lattice Polytopes by Their Volume

A well-known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends several existing ones, and has been used to correct mistakes in other classifications. When specialized to three-dimensional smooth polytopes, it extends previous classifications by Bogart et al., Lorenz, and Lundman. Moreover, we give a structure theorem for smooth polytopes with few lattice points that proves that they have a quadratic triangulation and which we use, together with the classification, to describe smooth polytopes having small volume in arbitrary dimension. In dimension three we enumerate all the simplices having up to 11 interior lattice points and we use them to conjecture a set of sharp inequalities for the coefficients of the Ehrhart $$h^*$$ -polynomials, unifying several existing conjectures. Finally, we extract and discuss some interesting minimal examples from the classification, and study the frequency of properties such as being spanning, very ample, IDP, and having a unimodular cover or triangulation. In particular, we find the smallest polytopes that are very ample but not IDP, and with a unimodular cover but without a unimodular triangulation.

Families of lattice polytopes of mixed degree one

It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension n≥4, showing that all but finitely many n-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex Δn−1. Furthermore, we give a complete solution in dimension n=3. In the course of this we show that our finiteness result does not extend to dimension n=3, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle Δ2.

Local optimality of Zaks-Perles-Wills simplices

In 1982, Zaks, Perles and Wills discovered a d-dimensional lattice simplex Sd,k with k interior lattice points, whose volume is linear in k and doubly exponential in the dimension d. It is conjectured that, for all d≥3 and k≥1, the simplex Sd,k is a volume maximizer in the family Pd(k) of all d-dimensional lattice polytopes with k interior lattice points. The simplex Sd,k has a facet with only one lattice point in the relative interior. We show that Sd,k is a volume maximizer in the family of simplices S∈Sd(k) that have a facet with one lattice point in its relative interior. We also show that, in the above family, the volume maximizer is unique up to unimodular transformations.

Stringy -functions of canonical toric Fano threefolds and their applications

Let be a -dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy -function of the -dimensional canonical toric Fano variety associated with . Using the stringy Libgober– Wood identity and our formula, we generalize the well-known combinatorial identity which holds for -dimensional reflexive polytopes .

Pattern-avoiding polytopes

Two well-known polytopes whose vertices are indexed by permutations in the symmetric group Sn are the permutohedron Pn and the Birkhoff polytope Bn. We consider polytopes Pn(Π) and Bn(Π), whose vertices correspond to the permutations in Sn avoiding a set of patterns Π. For various choices of Π, we explore the Ehrhart polynomials and h∗-vectors of these polytopes as well as other aspects of their combinatorial structure.For Pn(Π), we consider all subsets Π⊆S3 and are able to provide results in most cases. To illustrate, Pn(123,132) is a Pitman–Stanley polytope, the number of interior lattice points in Pn(132,312) is a derangement number, and the normalized volume of Pn(123,231,312) is the number of trees on n vertices.The polytopes Bn(Π) seem much more difficult to analyze, so we focus on four particular choices of Π. First we show that the Bn(231,321) is exactly the Chan–Robbins–Yuen polytope. Next we prove that for any Π containing {123,312} we have h∗(Bn(Π))=1. Finally, we study Bn(132,312) and B˜n(123), where the tilde indicates that we choose vertices corresponding to alternating permutations avoiding the pattern 123. In both cases we use order complexes of posets and techniques from toric algebra to construct regular, unimodular triangulations of the polytopes. The posets involved turn out to be isomorphic to the lattices of Young diagrams contained in a certain shape, and this permits us to give an exact expression for the normalized volumes of the corresponding polytopes via the hook formula. Finally, Stanley’s theory of (P,ω)-partitions allows us to show that their h∗-vectors are symmetric and unimodal.Various questions and conjectures are presented throughout.

Lattice Simplices with a Fixed Positive Number of Interior Lattice Points: A Nearly Optimal Volume Bound

Abstract We give an explicit upper bound on the volume of lattice simplices with fixed positive number of interior lattice points. The bound differs from the conjectural sharp upper bound only by a linear factor in the dimension. This improves significantly upon the previously best results by Pikhurko from 2001.

On Ehrhart Polynomials of Lattice Triangles

The Ehrhart polynomial of a lattice polygon $P$ is completely determined by the pair $(b(P),i(P))$ where $b(P)$ equals the number of lattice points on the boundary and $i(P)$ equals the number of interior lattice points. All possible pairs $(b(P),i(P))$ are completely described by a theorem due to Scott. In this note, we describe the shape of the set of pairs $(b(T),i(T))$ for lattice triangles $T$ by finding infinitely many new Scott-type inequalities.

On the n-th row of the graded Betti table of an n-dimensional toric variety

We prove an explicit formula for the first non-zero entry in the n-th row of the graded Betti table of an n-dimensional projective toric variety associated to a normal polytope with at least one interior lattice point. This applies to Veronese embeddings of projective space where we prove a special case of a conjecture of Ein and Lazarsfeld. We also prove an explicit formula for the entire n-th row when the interior of the polytope is one-dimensional. All results are valid over an arbitrary field k.

Intrinsicness of the Newton polygon for smooth curves on $${\mathbb {P}}^1\times {\mathbb {P}}^1$$ P 1 × P 1

Let C be a smooth projective curve in \({\mathbb {P}}^1 \times {\mathbb {P}}^1\) of genus \(g\ne 4\), and assume that it is birationally equivalent to a curve defined by a Laurent polynomial that is non-degenerate with respect to its Newton polygon \(\Delta \). Then we show that the convex hull \(\Delta ^{(1)}\) of the interior lattice points of \(\Delta \) is a standard rectangle, up to a unimodular transformation. Our main auxiliary result, which we believe to be interesting in its own right, is that the first scrollar Betti numbers of \(\Delta \)-non-degenerate curves are encoded in the combinatorics of \(\Delta ^{(1)}\), if \(\Delta \) satisfies some mild conditions.

A GENERALIZATION OF THE DISCRETE VERSION OF MINKOWSKI'S FUNDAMENTAL THEOREM

One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of an o-symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.