Projectively unique polytopes and toric slack ideals

Toric Ideals of Flow Polytopes

We show that toric ideals of flow polytopes are generated in degree 3. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gröbner bases of the toric ideal of the Birkhoff polytope Bn have at most degree n. We show that this bound is sharp fo...

Spin : a Software for Computing State Polytopes of Toric Ideals

Toric Ideals in Algebraic Statistics Toric Ideals in Algebraic Statistics

Toric Ideals in Algebraic Statistics

Toric Varieties and Lattice Polytopes

We begin with a lattice N isomorphic to Z. The dual lattice M of N is given by Hom(N,Z); it is also isomorphic to Z. (The alphabet may appear to be going backwards; but this notation is standard in the literature.) We write the pairing of v ∈ N and w ∈M as 〈v, w〉. A cone in N is a subset of the real vector space NR = N ⊗R generated by nonnegative R-linear combinations of a set of vectors {v1, ....

Toric Codes and Lattice Ideals

Let X be a complete simplicial toric variety over a finite field Fq with homogeneous coordinate ring S = Fq[x1, . . . , xr] and split torus TX ∼= (Fq). We prove that vanishing ideal of a subset Y of the torus TX is a lattice ideal if and only if Y is a subgroup. We show that these subgroups are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determin...