Geelen, Gerards, and Whittle [3] announced the following result: let $q =p^k$ be a prime power, and let $\mathcal{M}$ be a proper minor-closed class of$\mathrm{GF}(q)$-representable matroids, which does not contain$\mathrm{PG}(r-1,p)$ for sufficiently high $r$. There exist integers $k, t$such that every vertically $k$-connected matroid in $\mathcal{M}$ is arank-$(\leq t)$ perturbation of a frame matroid or the dual of a frame matroidover $\mathrm{GF}(q)$. They further announced a characterization of theperturbations through the introduction of subfield templates and frametemplates. We show a family of dyadic matroids that form a counterexample to thisresult. We offer several weaker conjectures to replace the ones in [3], discussconsequences for some published papers, and discuss the impact of these newconjectures on the structure of frame templates.
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机译:Geelen,Gerards和Whittle [3]宣布了以下结果:令$ q = p ^ k $为素数,令$ \ mathcal {M} $为$ \ mathrm {GF}的适当次要封闭类。 (q)个可表示的拟阵,其中不包含$ \ mathrm {PG}(r-1,p)$来表示足够高的$ r $。存在整数$ k,t $,使得$ \ mathcal {M} $中每个与$ k $垂直连接的拟阵均是框架拟阵或框架拟阵的对偶的arank-$(\ leq t)$扰动。 mathrm {GF}(q)$。他们还通过引入子场模板和框架模板来宣布扰动的特征。我们展示了一个二进制拟阵阵,构成了对此结果的反例。我们提供了几个较弱的猜想来代替[3]中的猜想,讨论了一些已发表论文的后果,并讨论了这些新猜想对框架模板结构的影响。
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