Syzygies over the Polytope Semiring
Abstract
Tropical geometry and its applications indicate a "theory of syzygies" over polytope semirings. Taking cue from this indication, we study a notion of syzygies over the polytope semiring. We begin our exploration with the concept of Newton basis, an analogue of Gröbner basis that captures the image of an ideal under the Newton polytope map. The image ${\rm New}(I)$ of a graded ideal $I$ under the Newton polytope is a graded subsemimodule of the polytope semiring. Analogous to the Hilbert series, we define the notion of NewtonHilbert series that encodes the rank of each graded piece of ${\rm New}(I)$. We prove the rationality of the NewtonHilbert series for subsemimodules that satisfy a property analogous to CohenMacaulayness. We define notions of regular sequence of polytopes and syzygies of polytopes. We show an analogue of the Koszul property characterizing the syzygies of a regular sequence of polytopes.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.07395
 Bibcode:
 2016arXiv160607395M
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 14T05;
 51M20
 EPrint:
 Major Revisions, To appear in the Journal of the London Mathematical Society