Extending Pappus' Theorem
Abstract
Let $M_1$ and $M_2$ be matroids such that $M_2$ arises from $M_1$ by relaxing a circuit-hyperplane.
We will prove that if $M_1$ and $M_2$ are both representable over some finite field $GF(q)$,
then $M_1$ and $M_2$ have highly structured representations.
Roughly speaking, $M_1$ and $M_2$ have representations that can be partitioned into a
bounded number of blocks each of which is \enquote{triangular}, a property we call weakly block-triangular.
Geelen, Gerards and Whittle have announced that, under the hypotheses above, the matroids
$M_1$ and $M_2$ both have pathwidth bounded by some constant depending only on $q$.
That result plays a significant role in their announced proof of Rota's Conjecture.
Bounding the pathwidth of $M_1$ and $M_2$ is currently the single most complicated part
in the proof of Rota's Conjecture. Our result is intended as a step toward simplifying this part.
A matroid $N$ is said to be a fragile minor of another matroid $M$ if $M/C\backslash D = N$ for
some $C,D \subseteq E(M)$, but $M/C'\backslash D' \neq N$ whenever $C \neq C'$ or $D \neq D'$.
As a second result, we will prove that, given a $GF(q)$-representable matroid $N$, every
$GF(q)$-representable matroid $M$ having $N$ as a fragile minor has a representation
which is weakly block-triangular.
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Cite this version of the work
Florian Hoersch
(2017).
Extending Pappus' Theorem. UWSpace.
http://hdl.handle.net/10012/12784
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