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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.05678 |
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| _version_ | 1866914243171844096 |
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| author | Jahangir, Rizwan |
| author_facet | Jahangir, Rizwan |
| contents | We propose a canonical local-to-global lattice theory for rational fans. We define the $\textit{ray lattice } L_{\mathrm{rays}}(Σ)$ and the $\textit{relation lattice } L_{\mathrm{rel}}(Σ)$ as invariants functorial under fan isomorphisms. We introduce $\textit{star-local relation lattices}$, defined via the relation lattice of the localized quotient fan, which capture the linear dependencies visible within local neighborhoods. We define a $\textit{codimension filtration}$ on the global relation lattice and prove a generation theorem: the global lattice is generated by local relations supported on the stars of cones of codimension at least 1. This filtration is sensitive to the facial structure of $Σ$; explicit examples and a conjecture suggest that subdivisions can only preserve or lower filtration depths, distinguishing fans with different combinatorial topologies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05678 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Canonical Lattices and Integer Relations Associated to Rational Fans Jahangir, Rizwan Combinatorics Algebraic Geometry We propose a canonical local-to-global lattice theory for rational fans. We define the $\textit{ray lattice } L_{\mathrm{rays}}(Σ)$ and the $\textit{relation lattice } L_{\mathrm{rel}}(Σ)$ as invariants functorial under fan isomorphisms. We introduce $\textit{star-local relation lattices}$, defined via the relation lattice of the localized quotient fan, which capture the linear dependencies visible within local neighborhoods. We define a $\textit{codimension filtration}$ on the global relation lattice and prove a generation theorem: the global lattice is generated by local relations supported on the stars of cones of codimension at least 1. This filtration is sensitive to the facial structure of $Σ$; explicit examples and a conjecture suggest that subdivisions can only preserve or lower filtration depths, distinguishing fans with different combinatorial topologies. |
| title | Canonical Lattices and Integer Relations Associated to Rational Fans |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2601.05678 |