37 research outputs found
Lattice polytopes in coding theory
In this paper we discuss combinatorial questions about lattice polytopes
motivated by recent results on minimum distance estimation for toric codes. We
also prove a new inductive bound for the minimum distance of generalized toric
codes. As an application, we give new formulas for the minimum distance of
generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure
Bezout Inequality for Mixed volumes
In this paper we consider the following analog of Bezout inequality for mixed
volumes: We show that the above
inequality is true when is an -dimensional simplex and are convex bodies in . We conjecture that if the above
inequality is true for all convex bodies , then must
be an -dimensional simplex. We prove that if the above inequality is true
for all convex bodies , then must be indecomposable
(i.e. cannot be written as the Minkowski sum of two convex bodies which are not
homothetic to ), which confirms the conjecture when is a
simple polytope and in the 2-dimensional case. Finally, we connect the
inequality to an inequality on the volume of orthogonal projections of convex
bodies as well as prove an isomorphic version of the inequality.Comment: 18 pages, 2 figures; an error in the isomorphic version of the
inequality is corrected (which improved the inequality
Toric Complete Intersection Codes
In this paper we construct evaluation codes on zero-dimensional complete intersections in toric varieties and give lower bounds for their minimum distance. This generalizes the results of Gold–Little–Schenck and Ballico–Fontanari who considered evaluation codes on complete intersections in the projective space
Classification of triples of lattice polytopes with a given mixed volume
We present an algorithm for the classification of triples of lattice
polytopes with a given mixed volume in dimension 3. It is known that the
classification can be reduced to the enumeration of so-called irreducible
triples, the number of which is finite for fixed . Following this algorithm,
we enumerate all irreducible triples of normalized mixed volume up to 4 that
are inclusion-maximal. This produces a classification of generic trivariate
sparse polynomial systems with up to 4 solutions in the complex torus, up to
monomial changes of variables. By a recent result of Esterov, this leads to a
description of all generic trivariate sparse polynomial systems that are
solvable by radicals
Generalized multiplicities of edge ideals
We explore connections between the generalized multiplicities of square-free
monomial ideals and the combinatorial structure of the underlying hypergraphs
using methods of commutative algebra and polyhedral geometry. For instance, we
show the -multiplicity is multiplicative over the connected components of a
hypergraph, and we explicitly relate the -multiplicity of the edge ideal of
a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of
its special fiber ring. In addition, we provide general bounds for the
generalized multiplicities of the edge ideals and compute these invariants for
classes of uniform hypergraphs.Comment: 24 pages, 6 figures. The results of Theorem 4.6 and Theorem 9.2 are
now more general. To appear in Journal of Algebraic Combinatoric
Wulff shapes and a characterization of simplices via a Bezout type inequality
Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii
we study the following Bezout type inequality for mixed volumes We show
that the above inequality characterizes simplices, i.e. if is a convex body
satisfying the inequality for all convex bodies , then must be an -dimensional simplex. The main idea of
the proof is to study perturbations given by Wulff shapes. In particular, we
prove a new theorem on differentiability of the support function of the Wulff
shape, which is of independent interest.
In addition, we study the Bezout inequality for mixed volumes introduced in
arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies
which is strictly larger than the set of all polytopes that are non-simplices.
We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly
indecomposable convex bodies
Tropical determinant on transportation polytope
Let be the set of all the integer points in the
transportation polytope of matrices with row sums and column
sums . In this paper we find the sharp lower bound on the tropical
determinant over the set . This integer
piecewise-linear programming problem in arbitrary dimension turns out to be
equivalent to an integer non-linear (in fact, quadratic) optimization problem
in dimension two. We also compute the sharp upper bound on a modification of
the tropical determinant, where the maximum over all the transversals in a
matrix is replaced with the minimum.Comment: 16 pages, 2 figure
Characterization of Simplices via the Bezout Inequality for Mixed volumes
We consider the following Bezout inequality for mixed volumes:
It was shown previously that
the inequality is true for any -dimensional simplex and any convex
bodies in . It was conjectured that simplices
are the only convex bodies for which the inequality holds for arbitrary bodies
in . In this paper we prove that this is indeed
the case if we assume that is a convex polytope. Thus the Bezout
inequality characterizes simplices in the class of convex -polytopes. In
addition, we show that if a body satisfies the Bezout inequality for
all bodies then the boundary of cannot have strict
points. In particular, it cannot have points with positive Gaussian curvature.Comment: 8 page
Eventual Quasi-Linearity of The Minkowski Length
The Minkowski length of a lattice polytope PP is a natural generalization of the lattice diameter of PP. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in PP. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates tPtP of a lattice polytope PP behaves polynomially in t∈Nt∈N. In this paper we prove that for any lattice polytope PP, the Minkowski length of tPtP for t∈Nt∈N is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of coordinates boxes, degree one polytopes, and dilates of unimodular simplices. In addition, we give a new bound for the Minkowski length of lattice polygons and show that the Minkowski length of a lattice triangle coincides with its lattice diameter
Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes
Let P1,..., Pn and Q1,...,Qn be convex polytopes in Rn such that Pi is a proper subset of Qi . It is well-known that the mixed volume has the monotonicity property: V (P1,...,Pn) is less than or equal to V (Q1,...,Qn) . We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P1,..., Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 U...U Pn . In addition, we obtain an analog of Cramer\u27s rule for sparse polynomial systems