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Syzygies over the Polytope Semiring
Combinatorics · Submitted 23 Jun 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show an analogue of the Koszul property characterizing the syzygies of a regular sequence of polytopes.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Tropical geometry and its applications indicate a "theory of syzygies" over polytope semirings. | background 1.00 | 0.05 | 0.08 | hedged possibility 0.97 |
| s1 | Taking cue from this indication, we study a notion of syzygies over the polytope semiring. | contribution 0.97 | 0.86 | 0.04 | asserted 0.92 |
| s2 | 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. | contribution 0.80 | 0.66 | 0.04 | asserted 0.95 |
| s3 | The image ${\rm New}(I)$ of a graded ideal $I$ under the Newton polytope is a graded sub-semimodule of the polytope semiring. | result 0.73 | 0.50 | 0.05 | asserted 1.00 |
| s4 | Analogous to the Hilbert series, we define the notion of Newton-Hilbert series that encodes the rank of each graded piece of ${\rm New}(I)$. | contribution 0.93 | 0.92 | 0.05 | asserted 0.99 |
| s5 | We prove the rationality of the Newton-Hilbert series for sub-semimodules that satisfy a property analogous to Cohen-Macaulayness. | result 0.99 | 0.95 | 0.22 | asserted 1.00 |
| s6 | We define notions of regular sequence of polytopes and syzygies of polytopes. | contribution 0.99 | 0.94 | 0.06 | asserted 0.94 |
| s7 | We show an analogue of the Koszul property characterizing the syzygies of a regular sequence of polytopes. | result 0.96 | 0.94 | 0.25 | asserted 1.00 |
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Original 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.
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Cites as background 5 papers
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Recent works of MacPherson [ref] , [ref] introduce an analogue of integral closure over idempotent semirings and a notion of projective modules over polyhedral semirings.
On the difference between `tropical functions' and real-valued functions · Motivation and Related Work · 0.98
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Recent works of MacPherson [ref] , [ref] introduce an analogue of integral closure over idempotent semirings and a notion of projective modules over polyhedral semirings.
Projective modules over polyhedral semirings · Motivation and Related Work · 0.98
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In a recent paper, Rowen [ref] introduces the concept of negation map on a semiring to handle the problem of the absence of additive inverses in a semiring.
Algebras with a negation map · Motivation and Related Work · 0.97
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