Syzygies over the Polytope Semiring

TL;DR

This paper develops a theory of syzygies over polytope semirings inspired by tropical geometry, introducing Newton bases, Newton-Hilbert series, and analogues of Cohen-Macaulayness and Koszul properties.

Contribution

It introduces the concept of syzygies over polytope semirings, including Newton bases and Newton-Hilbert series, and establishes their properties and analogues of classical algebraic concepts.

Findings

Rationality of Newton-Hilbert series for Cohen-Macaulay-like semimodules

Definition of regular sequences and syzygies of polytopes

Characterization of syzygies via an analogue of the Koszul property

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 Gr\"obner 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 sub-semimodule of the polytope semiring. Analogous to the Hilbert series, we define the notion of Newton-Hilbert series that encodes the rank of each graded piece of ${\rm New}(I)$. We prove the rationality of the Newton-Hilbert series for sub-semimodules that satisfy a property analogous to Cohen-Macaulayness. 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.