Convex Lattice Polygons Research Articles

Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics


 
 
 In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime $\ell$ for the function $p\big(n,m,N\big)$, which enumerates the partitions of $n$ into at most $m$ parts with no part larger than $N$. We also treat the function $p\big(n,m,(a,b]\big)$, which bounds the largest part above and below, and obtain similar infinite families of congruences.
 For $m \leq 4$ and $\ell = 3$, simple combinatorial statistics called "cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons.
 
 For $m \leq 4$ and $\ell = 3$, simple combinatorial statistics called ``cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons. 
 
 

Convex lattice polygons with all lattice points visible

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than 2, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves.

An extremal property of lattice polygons

We find the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given square sublattice. As a tool, we prove Diophantine inequalities relating the number of edges of a broken line and the coordinates of its endpoints.

An analogue of the Aleksandrov projection theorem for convex lattice polygons

An analogue of the Aleksandrov projection theorem for convex lattice polygons

Fock Spaces and Refined Severi Degrees

A convex lattice polygon Delta determines a pair (S,L) of a toric surface together with an ample toric line bundle on S. The Severi degree N^{Delta,delta} is the number of delta-nodal curves in the complete linear system |L| passing through dim|L|-delta general points. Cooper and Pandharipande showed that in the case of PP^1 x PP^1 the Severi degrees can be computed as the matrix elements of an operator on a Fock space. In this note we want to generalize and extend this result in two ways. First we show that it holds more generally for Delta a so called h-transverse lattice polygon. This includes the case of PP^2 and rational ruled surfaces, but also many other, also singular, surfaces. Using a deformed version of the Heisenberg algebra, we extend the result to the refined Severi degrees defined and studied by G\ottsche and Shende and by Block and G\ottsche. For Delta an h-transverse lattice polygon, one can, following Brugall\'e and Mikhalkin, replace the count of tropical curves by a count of marked floor diagrams, which are slightly simpler combinatorial objects. We show that these floor diagrams are the Feynman diagrams of certain operators on a Fock space, proving the result.

Lattice polygons with two interior lattice points

A lattice point in the plane is a point with integer coordinates. A lattice segment is a line segment whose endpoints are lattice points. A lattice polygon is a simple polygon whose vertices are lattice points. We find all convex lattice polygons in the plane up to equivalence with two interior lattice points.

On the classification of convex lattice polytopes

Abstract In 1980, Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its high dimensional analogue have been studied by Bárány, Pach, Vershik and others. Bounds for the number of non-equivalent d-dimensional convex lattice polytopes of given volume have been achieved. In this paper we study Arnold's problem for centrally symmetric lattice polygons and the classification problem for convex lattice polytopes of given cardinality. In the plane we obtain analogues to the bounds of Arnold, Bárány and Pach in both cases. However, the number of non-equivalent d-dimensional convex lattice polytopes of w lattice points is infinite whenever w – 1 ≥ d ≥ 3, which may intuitively contradict to Bárány and Vershik's upper bound.

The Newton Polygon of a Rational Plane Curve

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kušnirenko–Bernštein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic rational functions, with explicit genericity conditions. We also show that the variety of rational curves with given Newton polygon is unirational and we compute its dimension. As a consequence, we obtain that any convex lattice polygon with positive area is the Newton polygon of a rational plane curve.

On the combinatorial classification of toric log del Pezzo surfaces

AbstractToric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. Upper bounds on the volume and on the number of boundary lattice points of these polygons are derived in terms of the indexℓ. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for allℓ≤16 is obtained.

Lattice Polygons and the Number 2<I>i</I> + 7

In this note we classify all triples (a,b,i) such that there is a convex lattice polygon P with area a, and b respectively i lattice points on the boundary respectively in the interior. The crucial lemma for the classification is the necessity of b \le 2 i + 7. We sketch three proofs of this fact: the original one by Scott, an elementary one, and one using algebraic geometry. As a refinement, we introduce an onion skin parameter l: how many nested polygons does P contain? and give sharper bounds.

Toric Surface Codes and Minkowski Length of Polygons

In this paper we prove new lower bounds for the minimum distance of a toric surface code $\mathcal{C}_P$ defined by a convex lattice polygon $P\subset\mathbb{R}^2$. The bounds involve a geometric invariant $L(P)$, called the full Minkowski length of P. We also show how to compute $L(P)$ in polynomial time in the number of lattice points in P.

Algebraic computations and applications to geometry (abstract only)

Real algebraic numbers are the real numbers that are real roots of univariate polynomials with integer coefficients. We study exact algorithms, from a theoretical and an implementation point of view, based on integer arithmetic of arbitrary precision, for computations with real algebraic numbers and applications of these algorithms on problems and algorithms in non linear computational geometry. In order to construct a real algebraic number we must compute the real roots of a univariate polynomial with integer cofficients. We unify and simplify the theory behind the subdivision based algorithms for real root isolation and we improve the complexity of the algorithm that is based on the continued fraction expansion of the real numbers. The best known complexity bound up today is achieved using new techniques. Moreover, we prove that the bound holds for non square-free polynomials and that in the same complexity bound we can compute the multiplicities of the real roots. We prove a new bound for the expected complexity of the algorithm based on continued fractions. We generalize the real root isolation algorithms to bivariate polynomial systems. Our experimental analysis proves the effectiveness of our methods. The algorithms that we consider for computations with real algebraic numbers are construction, comparison, sign evaluation and quantifier elimination. If the degree of the polynomial is small, i.e. ≤ 4 in the univariate case and ≤ 2 in the bivariate case, we propose special purpose algorithms that have constant arithmetic complexity. For all the algorithms we present a C++ implementation and an experimental analysis. In computational geometry we study the predicates needed by the algorithms for the arrangement of elliptic arcs in the plane and the computation of the Voronoi diagram of ellipses, also in the plane. Finally, given a convex lattice polygon we study algorithms for decomposing it to two other convex lattice polygons, such that their Minkowski sum is the original polygon.

Elementary notions of lattice trigonometry

In this paper we study properties of lattice trigonometric functions of lattice angles in lattice geometry. We introduce the definition of sums of lattice angles and establish a necessary and sufficient condition for three angles to be the angles of some lattice triangle in terms of lattice tangents. This condition is a version of the Euclidean condition: three angles are the angles of some triangle iff their sum equals \pi. Further we find the necessary and sufficient condition for an ordered n-tuple of angles to be the angles of some convex lattice polygon. In conclusion we show applications to theory of complex projective toric varieties, and a list of unsolved problems and questions.

On maximal convex lattice polygons inscribed in a plane convex set

Given a convex compact setK ⊂ ℝ2 what is the largestn such thatK contains a convex latticen-gon? We answer this question asymptotically. It turns out that the maximaln is related to the largest affine perimeter that a convex set contained inK can have. This, in turn, gives a new characterization ofK 0, the convex set inK having maximal affine perimeter.

A proof of Coleman's conjecture

Almost thirty years ago Coleman made a conjecture that for any convex lattice polygon with v vertices, g ( g ⩾ 1 ) interior lattice points and b boundary lattice points we have b ⩽ 2 g - v + 10 . In this note we give a proof of the conjecture. We also aim to describe all convex lattice polygons for which the bound b = 2 g - v + 10 is attained.

A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces

The Welschinger invariants of real rational algebraic surfaces are natural analogues of the genus zero Gromov-Witten invariants. We establish a tropical formula to calculate the Welschinger invariants of real toric Del Pezzo surfaces for any conjugation-invariant configuration of points. The formula expresses the Welschinger invariants via the total multiplicity of certain tropical curves (non-Archimedean amoebas) passing through generic configurations of points, and then via the total multiplicity of some lattice paths in the convex lattice polygon associated with a given surface. We also present the results of computation of Welschinger invariants, obtained jointly with I. Itenberg and V. Kharlamov.

Lattice Polygons in the Plane and the Number 12

A convex polygon P in R all of whose vertices have integer coordinates is called a convex lattice polygon. If the polygon has n lattice points on its boundary, represented by the vectors p1, . . . , pn (in anticlockwise order), then we say that the length l(P) is n. The dual (convex) lattice polygon P∨ is by definition the convex hull of the difference vectors qi = pi+1 − pi, where indices throughout the note are considered modulo n. In this note we will give a simple proof of the fact that if P be a convex lattice polygon in R whose only interior lattice point is the origin, then l(P) + l(P∨) = 12. This result has its origin in a correspondence between convex lattice polygons and certain toric varieties, and has several ingenious proofs (see [1] and [2]). These proofs use either Noether’s formula or modular forms to explain the occurrence of the number 12. Our proof observes that if l(P) increases by one then l(P∨) decreases by one, so that their sum is constant. The number 12 then appears as 3 (the smallest possible length) plus 9 (the largest possible length.) Our proof (which grew out of the second authors project, supervised by the first author) also supports the suggestion in [2] that the number 3 can be viewed as a discrete analogue of π in this context.

Limit shape of convex lattice polygons with minimal perimeter

Let n be a positive integer and ∥ · ∥ any norm in R 2 . Denote by B the unit ball of ∥ · ∥ and P B , n the class of convex lattice polygons with n vertices and least ∥ · ∥ -perimeter. We prove that after suitable normalization, all members of P B , n tend to a fixed convex body, as n → ∞ .

Convex lattice polygons of fixed area with perimeter-dependent weights

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight tm to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s(-theta(conv))eK(t)square root(s) for large s and t less than a critical threshold tc, where K(t) is a t-dependent constant, and theta(conv) is a critical exponent which does not change with t. Using heuristic arguments, we find that theta(conv) is 1/4 for the square lattice, but -1/4 for the hexagonal lattice. The reason for this unexpected nonuniversality of theta(conv) is traced to existence of sharp corners in the asymptotic shape of these polygons.

Simplification of surface parametrizations—a lattice polygon approach

Given a convex lattice polygon, we compute a descending sequence of lattice polygons obtained by repeatedly passing to the convex hull of the interior lattice points. This process gives the idea for an algorithm that simplifies a given parametric surface by reparametrization.