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Subdivisions of toric complexes

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Morten Brun, Tim Roemer

Commutative Algebra · Submitted 22 Jan 2004

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels.

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arXiv IDmath/0401292 (version 3)
Pith ID77RTR36R, Pith's permanent record of this paper and its review
CitationMorten Brun, Tim Roemer. "Subdivisions of toric complexes." arXiv:math/0401292 (2004). https://pith.science/paper/77RTR36R
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,668 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.86 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extend Stanley and Reisner's face ring for abstract simplicial complexes and Stanley's face ring for rational fans. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels. We apply our results to study edgewise subdivisions of abstract simplicial complexes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. contribution 1.00 0.93 0.09 asserted 1.00
s1 Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extend Stanley and Reisner's face ring for abstract simplicial complexes and Stanley's face ring for rational fans. background 0.70 0.28 0.11 asserted 0.98
s2 Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels. result 0.67 0.92 0.59 asserted 1.00
s3 We apply our results to study edgewise subdivisions of abstract simplicial complexes. contribution 0.97 0.89 0.12 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extend Stanley and Reisner's face ring for abstract simplicial complexes and Stanley's face ring for rational fans. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels. We apply our results to study edgewise subdivisions of abstract simplicial complexes.

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