97,332 research outputs found

### A note on lattice-face polytopes and their Ehrhart polynomials

We give a new definition of lattice-face polytopes by removing an unnecessary
restriction in the paper "Ehrhart polynomials of lattice-face polytopes", and
show that with the new definition, the Ehrhart polynomial of a lattice-face
polytope still has the property that each coefficient is the normalized volume
of a projection of the original polytope. Furthermore, we show that the new
family of lattice-face polytopes contains all possible combinatorial types of
rational polytopes.Comment: 11 page

### Moduli of Crude Limit Linear Series

Eisenbud and Harris introduced the theory of limit linear series, and
constructed a space parametrizing their limit linear series. Recently, Osserman
introduced a new space which compactifies the Eisenbud-Harris construction. In
the Eisenbud-Harris space, the set of refined limit linear series is always
dense on a general reducible curve. Osserman asks when the same is true for his
space. In this paper, we answer his question by characterizing the situations
when the crude limit linear series contain an open subset of his space.Comment: 13 page

### Combinatorial bases for multilinear parts of free algebras with double compatible brackets

Let X be an ordered alphabet. Lie_2(n) (and P_2(n) respectively) are the
multilinear parts of the free Lie algebra (and the free Poisson algebra
respectively) on X with a pair of compatible Lie brackets. In this paper, we
prove the dimension formulas for these two algebras conjectured by B. Feigin by
constructing bases for Lie_2(n) (and P_2(n)) from combinatorial objects. We
also define a complementary space Eil_2(n) to Lie_2(n), give a pairing between
Lie_2(n) and Eil_2(n), and show that the pairing is perfect.Comment: 38 pages; 10 figure

### A combinatorial analysis of Severi degrees

Based on results by Brugall\'e and Mikhalkin, Fomin and Mikhalkin give
formulas for computing classical Severi degrees $N^{d, \delta}$ using long-edge
graphs. In 2012, Block, Colley and Kennedy considered the logarithmic version
of a special function associated to long-edge graphs appeared in
Fomin-Mikhalkin's formula, and conjectured it to be linear. They have since
proved their conjecture. At the same time, motivated by their conjecture, we
consider a special multivariate function associated to long-edge graphs that
generalizes their function. The main result of this paper is that the
multivariate function we define is always linear. A special case of our result
gives an independent proof of Block-Colley-Kennedy's conjecture.
The first application of our linearity result is that by applying it to
classical Severi degrees, we recover quadraticity of $Q^{d, \delta}$ and a
bound $\delta$ for the threshold of polynomiality of $N^{d, \delta}.$ Next, in
joint work with Osserman, we apply the linearity result to a special family of
toric surfaces and obtain universal polynomial results having connections to
the G\"ottsche-Yau-Zaslow formula. As a result, we provide combinatorial
formulas for the two unidentified power series $B_1(q)$ and $B_2(q)$ appearing
in the G\"ottsche-Yau-Zaslow formula.
The proof of our linearity result is completely combinatorial. We define
$\tau$-graphs which generalize long-edge graphs, and a closely related family
of combinatorial objects we call $(\tau, n)$-words. By introducing height
functions and a concept of irreducibility, we describe ways to decompose
certain families of $(\tau, n)$-words into irreducible words, which leads to
the desired results.Comment: 38 pages, 1 figure, 1 table. Major revision: generalized main results
in previous version. The old results only applies to classical Severi
degrees. The current version also applies to Severi degrees coming from
special families of toric surface

### Higher integrality conditions, volumes and Ehrhart polynomials

A polytope is integral if all of its vertices are lattice points. The
constant term of the Ehrhart polynomial of an integral polytope is known to be
1. In previous work, we showed that the coefficients of the Ehrhart polynomial
of a lattice-face polytope are volumes of projections of the polytope. We
generalize both results by introducing a notion of $k$-integral polytopes,
where 0-integral is equivalent to integral. We show that the Ehrhart polynomial
of a $k$-integral polytope $P$ has the properties that the coefficients in
degrees less than or equal to $k$ are determined by a projection of $P$, and
the coefficients in higher degrees are determined by slices of $P$. A key step
of the proof is that under certain generality conditions, the volume of a
polytope is equal to the sum of volumes of slices of the polytope.Comment: 30 pages, 1 figur

### Perturbation of transportation polytopes

We describe a perturbation method that can be used to reduce the problem of
finding the multivariate generating function (MGF) of a non-simple polytope to
computing the MGF of simple polytopes. We then construct a perturbation that
works for any transportation polytope. We apply this perturbation to the family
of central transportation polytopes of order kn x n, and obtain formulas for
the MGFs of the feasible cone of each vertex of the polytope and the MGF of the
polytope. The formulas we obtain are enumerated by combinatorial objects. A
special case of the formulas recovers the results on Birkhoff polytopes given
by the author and De Loera and Yoshida. We also recover the formula for the
number of maximum vertices of transportation polytopes of order kn x n.Comment: 25 pages, 3 figures. To appear in Journal of Combinatorial Theory
Ser.

### Ehrhart polynomials of cyclic polytopes

The Ehrhart polynomial of an integral convex polytope counts the number of
lattice points in dilates of the polytope. In math.CO/0402148, the authors
conjectured that for any cyclic polytope with integral parameters, the Ehrhart
polynomial of it is equal to its volume plus the Ehrhart polynomial of its
lower envelope and proved the case when the dimension d = 2. In our article, we
prove the conjecture for any dimension.Comment: 15 page

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