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 @math of a graded ideal @math 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 @math . 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.
Polytope semirings (under the name polytope algebra) appeared in the book of Pachter and Sturmfels [Chapter 2] PacStu05 in the context of computational biology. Semiring theory, in particular idempotent semiring theory, has been treated in several books and articles, see for example @cite_12 . Linear algebra over semirings has been the focus of these works.
@cite_12
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Preface. 1. Hemirings and semirings: definitions and examples. 2. Sets and relations with values in a semiring. 3. Building new semirings from old. 4. Some conditions on semirings. 5. Complemented elements in semirings. 6. Ideals in semirings. 7. Prime and semiprime ideals in semirings. 8. Factor semirings. 9. Morphisms of semirings. 10. Kernels of morphisms. 11. Semirings of fractions. 12. Euclidean semirings. 13. Additively-regular semirings. 14. Semimodules over semirings. 15. Factor semimodules. 16. Some constructions for semimodules. 17. Free, projective, and injective semimodules. 18. Localization of semimodules. 19. Linear algebra over a semiring. 20. Partially-ordered semirings. 21. Lattice-ordered semirings. 22. Complete semirings. 23. Complete semimodules. 24. CLO-semirings. 25. Fixed points of affine maps. References. Index of applications. Index of terminology.