复杂可表示矩阵的西尔维斯特-加莱类型定理

Pub Date : 2024-06-11 DOI:10.1007/s00454-024-00661-x

Jim Geelen, Matthew E. Kroeker

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引用次数: 0

摘要

西尔维斯特-加莱定理（Sylvester-Gallai Theorem）指出，每个秩为 3 的实可表示 matroid 都有一条两点线。我们证明，对于每一个 \(kge 2\), 每一个秩至少为 \(4^{k-1}\) 的复可表示 matroid 都有一个正好有 k 个点的 rank-k 平面。对于 \(k=2\) 来说，这是凯利（Kelly）提出的一个著名结果，我们在证明中使用了这个结果。类似的结果早先由巴拉克、德维尔、维格德森和耶胡达约夫证明，后来由德维尔、萨拉夫和维格德森完善，但我们用更基本的证明得到了更好的边界。

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A Sylvester–Gallai-Type Theorem for Complex-Representable Matroids

The Sylvester–Gallai Theorem states that every rank-3 real-representable matroid has a two-point line. We prove that, for each \(k\ge 2\), every complex-representable matroid with rank at least \(4^{k-1}\) has a rank-k flat with exactly k points. For \(k=2\), this is a well-known result due to Kelly, which we use in our proof. A similar result was proved earlier by Barak, Dvir, Wigderson, and Yehudayoff and later refined by Dvir, Saraf, and Wigderson, but we get slightly better bounds with a more elementary proof.

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