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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.02826 |
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| _version_ | 1866917603371384832 |
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| author | Campbell, Rutger Geelen, Jim Kroeker, Matthew E. |
| author_facet | Campbell, Rutger Geelen, Jim Kroeker, Matthew E. |
| contents | Melchior's inequality implies that the average line-length in a simple, rank-$3$, real-representable matroid is less than $3$. A similar result holds for complex-representable matroids, using Hirzebruch's inequality, but with a weaker bound of $4$. We show that the average plane-size in a simple, rank-$4$, complex-representable matroid is bounded above by an absolute constant, unless the matroid is the direct-sum of two lines. We also prove that, for any integer $k$, in complex-representable matroids with rank at least $2k-1$, the average size of a rank-$k$ flat is bounded above by a constant depending only on $k$. Finally, we prove that, for any integer $r\ge 2$, the average flat-size in rank-$r$ complex-representable matroids is bounded above by a constant depending only on $r$. We obtain our results using a theorem, due to Ben Lund, that gives a good estimate on the number of rank-$k$ flats in a complex-representable matroid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02826 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Average plane-size in complex-representable matroids Campbell, Rutger Geelen, Jim Kroeker, Matthew E. Combinatorics Melchior's inequality implies that the average line-length in a simple, rank-$3$, real-representable matroid is less than $3$. A similar result holds for complex-representable matroids, using Hirzebruch's inequality, but with a weaker bound of $4$. We show that the average plane-size in a simple, rank-$4$, complex-representable matroid is bounded above by an absolute constant, unless the matroid is the direct-sum of two lines. We also prove that, for any integer $k$, in complex-representable matroids with rank at least $2k-1$, the average size of a rank-$k$ flat is bounded above by a constant depending only on $k$. Finally, we prove that, for any integer $r\ge 2$, the average flat-size in rank-$r$ complex-representable matroids is bounded above by a constant depending only on $r$. We obtain our results using a theorem, due to Ben Lund, that gives a good estimate on the number of rank-$k$ flats in a complex-representable matroid. |
| title | Average plane-size in complex-representable matroids |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.02826 |