Average plane-size in complex-representable matroids

TL;DR

This paper establishes bounds on the average size of flats in complex-representable matroids, extending known inequalities and providing new universal bounds depending only on rank, with implications for combinatorial geometry.

Contribution

It proves new universal bounds on average flat sizes in complex-representable matroids, generalizing previous results and using advanced combinatorial estimates.

Findings

Average plane-size in rank-4 complex matroids is bounded unless the matroid is a direct sum of two lines.

For rank at least 2k-1, average size of rank-k flats is bounded by a constant depending only on k.

Average flat-size in rank-r complex matroids is bounded by a constant depending only on r.

Abstract

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.