9 research outputs found
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
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
Overdetermined Systems of Equations on Toric, Spherical, and Other Algebraic Varieties
Let be a collection of linear series on an algebraic variety
over . That is, is a finite
dimensional subspace of the space of regular sections of line bundles . Such a collection is called overdetermined if the generic
system with does not have any roots
on . In this paper we study solvable systems which are given by an
overdetermined collection of linear series. Generalizing the notion of a
resultant hypersurface we define a consistency variety as the closure of the set of all systems which have at least one common
root and study general properties of zero sets of a generic
consistent system . Then, in the case of equivariant linear
series on spherical homogeneous spaces we provide a strategy for computing
discrete invariants of such generic non-empty set . For equivariant
linear series on the torus this strategy provides explicit
calculations and generalizes the theory of Newton polyhedra.Comment: Improved exposition, minor changes. 18 pages, comments are welcome
Inequalities between mixed volumes of convex bodies: volume bounds for the Minkowski sum
In the course of classifying generic sparse polynomial systems which are
solvable in radicals, Esterov recently showed that the volume of the Minkowski
sum of -dimensional lattice polytopes is bounded from above
by a function of order , where is the mixed volume of the tuple
. This is a consequence of the well-known Aleksandrov-Fenchel
inequality. Esterov also posed the problem of determining a sharper bound. We
show how additional relations between mixed volumes can be employed to improve
the bound to , which is asymptotically sharp. We furthermore prove a
sharp exact upper bound in dimensions 2 and 3. Our results generalize to tuples
of arbitrary convex bodies with volume at least one.Comment: 21 pages, 5 figures; error in the statement of Theorem 3.3 correcte
Recommended from our members
Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications
Lattice polytopes arise naturally in many different branches of pure and applied mathematics such as number theory, commutative algebra, combinatorics, toric geometry, optimization, and mirror symmetry. The miniworkshop on “Lattice polytopes: methods, advances, applications” focused on two current hot topics: the classification of lattice polytopes with few lattice points and unimodality questions for Ehrhart polynomials. The workshop consisted of morning talks on recent breakthroughs and new methods, and afternoon discussion groups where participants from a variety of different backgrounds explored further applications, identified open questions and future research directions, discussed specific examples and conjectures, and collaboratively tackled open research problems