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.
Recent works of MacPherson @cite_4 , @cite_7 introduce an analogue of integral closure over idempotent semirings and a notion of projective modules over polyhedral semirings. Our work is another step towards commutative algebra over semirings. Our results are over polytope semirings and do not seem to directly extend to arbitrary semirings. This is primarily because of our use of the Lebseque measure on the set of polytopes. Other contexts where polytope semirings appear are Litvinov @cite_14 , Connes [Page 19] Con15 . Litvinov @cite_14 emphasizes a correspondence principle between classical analysis and idempotent analysis. According to this principle, every result in classical analysis has an idempotent analogue. Theorem and Theorem confirm a similar interplay between commutative rings and polytope semirings.
@cite_14 @cite_4 @cite_7
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A brief introduction to tropical and idempotent mathematics (with an emphasys on idempotent functional analysis) is presented. Applications to classical mechanics and geometry are especially examined. I classify projective modules over idempotent semirings that are free on a monoid. The analysis extends to the case of the semiring of convex, piecewise-affine functions on a polyhedron, for which projective modules correspond to convex families of weight polyhedra for the general linear group. I introduce the concept of integral closure for elements and ideals in idempotent semirings, and establish how it corresponds to its namesake in commutative algebra. In the case of free semirings, integral closure can be understood in terms of a certain monoid of convex bodies under Minkowski sum. I argue that integral closure therefore accounts for the difference between tropical functions' and real functions.