Partition Matrices for Polytopes towards Computing Toric Residue

Combinatorial Construction of Toric Residues

The toric residue is a map depending on n + 1 divisors on a complete toric variety of dimension n . It appears in a variety of contexts such as sparse polynomial systems, mirror symmetry, and GKZ hypergeometric functions. In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated po...

Quadratic Gröbner bases for smooth 3 × 3 transportation polytopes

The toric ideals of 3 × 3 transportation polytopes Trc are quadratically generated. The only exception is the Birkhoff polytope B3. If Trc is not a multiple of B3, these ideals even have square-free quadratic initial ideals. This class contains all smooth 3 × 3 transportation polytopes.

Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants

Resultants are defined in the sparse (or toric) context in order to exploit the structure of the polynomials as expressed by their Newton polytopes. Since determinantal formulae are not always possible, the most efficient general method for computing resultants is as the ratio of two determinants. This is made possible by Macaulay’s seminal result [15] in the dense homogeneous case, extended by...

Toric Residue Mirror Conjecture for Calabi-yau Complete Intersections

The toric residue mirror conjecture of Batyrev and Materov [2] for Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties expresses a toric residue as a power series whose coefficients are certain integrals over moduli spaces. This conjecture was proved independently by Szenes and Vergne [10] and Borisov [5]. We build on the work of these authors to generalize the residue mirror map to not...

Spin : a Software for Computing State Polytopes of Toric Ideals