Superbinomial coefficients

We investigate several families of polynomials that are related to certain Euler type summation operators. Being integer valued at integral points, they satisfy combinatorial properties and nearby symmetries, due to triangle recursion relations involving squares of polynomials. Some of these interpolate the Delannoy numbers. The results are motivated by and strongly related to our study of irreducible Lie supermodules, where dimension polynomials in many cases show similar features.

Matroids, Cyclic Flats, and Polyhedra

<p>Matroids have a wide variety of distinct, cryptomorphic axiom systems that are capable of defining them. A common feature of these is that they are able to be efficiently tested, certifying whether a given input complies with such an axiom system in polynomial time. Joseph Bonin and Anna de Mier, rediscovering a theorem first proved by Julie Sims, developed an axiom system for matroids in terms of their cyclic flats and the ranks of those cyclic flats. As with other matroid axiom systems, this is able to be tested in polynomial time. Distinct, non-isomorphic matroids may each have the same lattice of cyclic flats, and so matroids cannot be defined solely in terms of their cyclic flats. We do not have a clean characterisation of families of sets that are cyclic flats of matroids. However, it may be possible to tell in polynomial time whether there is any matroid that has a given lattice of subsets as its cyclic flats. We use Bonin and de Mier’s cyclic flat axioms to reduce the problem to a linear program, and show that determining whether a given lattice is the lattice of cyclic flats of any matroid corresponds to finding integral points in the solution space of this program, these points representing the possible ranks that may be assigned to the cyclic flats. We distinguish several classes of lattice for which solutions may be efficiently found, based upon the nature of the matrix of coefficients of the linear program, and of the polyhedron it defines, and then identify families of lattice that belong to those classes. We define operations and transformations on lattices of sets by examining matroid operations, and examine how these operations affect membership in the aforementioned classes. We conjecture that it is always possible to determine, in polynomial time, whether a given collection of subsets makes up the lattice of cyclic flats of any matroid.</p>

Matroids, Cyclic Flats, and Polyhedra

<p>Matroids have a wide variety of distinct, cryptomorphic axiom systems that are capable of defining them. A common feature of these is that they are able to be efficiently tested, certifying whether a given input complies with such an axiom system in polynomial time. Joseph Bonin and Anna de Mier, rediscovering a theorem first proved by Julie Sims, developed an axiom system for matroids in terms of their cyclic flats and the ranks of those cyclic flats. As with other matroid axiom systems, this is able to be tested in polynomial time. Distinct, non-isomorphic matroids may each have the same lattice of cyclic flats, and so matroids cannot be defined solely in terms of their cyclic flats. We do not have a clean characterisation of families of sets that are cyclic flats of matroids. However, it may be possible to tell in polynomial time whether there is any matroid that has a given lattice of subsets as its cyclic flats. We use Bonin and de Mier’s cyclic flat axioms to reduce the problem to a linear program, and show that determining whether a given lattice is the lattice of cyclic flats of any matroid corresponds to finding integral points in the solution space of this program, these points representing the possible ranks that may be assigned to the cyclic flats. We distinguish several classes of lattice for which solutions may be efficiently found, based upon the nature of the matrix of coefficients of the linear program, and of the polyhedron it defines, and then identify families of lattice that belong to those classes. We define operations and transformations on lattices of sets by examining matroid operations, and examine how these operations affect membership in the aforementioned classes. We conjecture that it is always possible to determine, in polynomial time, whether a given collection of subsets makes up the lattice of cyclic flats of any matroid.</p>

On the torsion values for sections of an elliptic scheme

We shall consider sections of a complex elliptic scheme ℰ {{{\mathcal{E}}}} over an affine base curve B, and study the points of B where the section takes a torsion value. In particular, we shall relate the distribution in B of these points with the canonical height of the section, proving an integral formula involving a measure on B coming from the so-called Betti map of the section. We shall show that this measure is the same one which appears in dynamical issues related to the section. This analysis will also involve the multiplicity with which a torsion value is attained, which is an independent problem. We shall prove finiteness theorems for the points where the multiplicity is higher than expected. Such multiplicity has also a relation with Diophantine Approximation and quasi-integral points on ℰ {{{\mathcal{E}}}} (over the affine ring of B), and in Sections 5 and 6 of the paper we shall exploit this viewpoint, proving an effective result in the spirit of Siegel’s theorem on integral points.

On some estimates and topological properties of relative orbits of subsets

In this paper, we give some topological properties and estimates of orbit of certain subsets of [Formula: see text]-points of varieties under actions of algebraic tori. These results are concerned with an analogue of Bruhat-Tits’ question on the set of [Formula: see text]-adic integral points of algebraic tori.

Projective toric codes

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.

Compactifications of Cluster Varieties and Convexity

Gross–Hacking–Keel–Kontsevich [13] discuss compactifications of cluster varieties from positive subsets in the real tropicalization of the mirror. To be more precise, let ${\mathfrak{D}}$ be the scattering diagram of a cluster variety $V$ (of either type– ${\mathcal{A}}$ or ${\mathcal{X}}$), and let $S$ be a closed subset of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$—the ambient space of ${\mathfrak{D}}$. The set $S$ is positive if the theta functions corresponding to the integral points of $S$ and its ${\mathbb{N}}$-dilations define an ${\mathbb{N}}$-graded subalgebra of $\Gamma (V, \mathcal{O}_V){ [x]}$. In particular, a positive set $S$ defines a compactification of $V$ through a Proj construction applied to the corresponding ${\mathbb{N}}$-graded algebra. In this paper, we give a natural convexity notion for subsets of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$, called broken line convexity, and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$ or to check positivity of a given subset.