We relate two conjectures that play a central role in the reported proof of Rota's Conjecture. Let $\mathbb{F}$ be a finite field. The first conjecture states that: the branch-width of any $\mathbb{F}$-representable N-fragile matroid is bounded by a function depending only upon $\mathbb{F}$ and N. The second conjecture states that: if a matroid $M_2$ is obtained from a matroid $M_1$ by relaxing a circuit-hyperplane and both $M_1$ and $M_2$ are $\mathbb{F}$-representable, then the branch-width of $M_1$ is bounded by a function depending only upon $\mathbb{F}$. Our main result is that the second conjecture implies the first.

中文翻译：

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Matroid的脆弱性和电路超平面的松弛

我们关联了两个猜想，它们在报道的Rota猜想的证明中起着核心作用。让$\mathbb{F}$是一个有限域。第一个猜想指出：$\mathbb{F}$可表示的N-脆弱拟阵受函数限制，仅取决于$\mathbb{F}$和Ñ。第二个猜想指出：如果拟阵${\mathrm{中号}}_2$ 从拟阵获得 ${\mathrm{中号}}_{\mathrm{1个}}$ 通过放宽电路超平面 ${\mathrm{中号}}_{\mathrm{1个}}$ 和 ${\mathrm{中号}}_2$ 是 $\mathbb{F}$-representable，然后是 ${\mathrm{中号}}_{\mathrm{1个}}$ 受一个函数的限制，仅取决于 $\mathbb{F}$。我们的主要结果是第二个猜想隐含着第一个猜想。