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Syzygies over the polytope semiring
Author(s) -
Manjunath Madhusudan
Publication year - 2017
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12065
Subject(s) - polytope , mathematics , semiring , hilbert–poincaré series , ideal (ethics) , rank (graph theory) , series (stratigraphy) , hilbert series and hilbert polynomial , sequence (biology) , combinatorics , birkhoff polytope , property (philosophy) , algebra over a field , discrete mathematics , hilbert space , pure mathematics , geometry , paleontology , philosophy , convex set , genetics , epistemology , convex optimization , regular polygon , biology
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 imageNew ( I ) of a graded ideal I under the Newton polytope map 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 ofNew ( I ) . We prove the rationality of the Newton–Hilbert series for sub‐semimodules that satisfy a property analogous to Cohen‐Macaulayness. We define the 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.