Average plane-size in complex-representable matroids

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发表日期：

2025年10月2日

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Rutger Campbell,
Jim Geelen,
Matthew E. Kroeker

摘要

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 $2k-1$ $2 k - 1$ , 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 $r\geqslant 2$ $r ⩾ 2$ , 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.