Vámos 拟阵没有第二对称幂
The Vámos Matroid Has No Second Symmetric Power
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中文总结 AI 辅助
本文证明 Vámos 拟阵 $V_8$ 没有第二对称幂，从而否定地回答了 Anderson 的问题，并由此得出其相关热带线性空间不能由热带理想实现。
中文摘要 AI 辅助
拟阵的对称幂由 Lovász 和 Mason 引入。该主题近来因与热带理想实现热带线性空间之间的联系而重新引起关注。Anderson 证明了与拟阵 $M$ 相关的热带线性空间是某个热带理想的簇，当且仅当 $M$ 对每个正整数 $d$ 都有第 $d$ 对称幂。他还提出了 Vámos 拟阵 $V_8$ 是否有第二对称幂的问题。我们通过证明 $V_8$ 没有第二对称幂来否定地回答该问题。根据 Anderson 的刻画，相关的热带线性空间不能被实现为热带理想的簇。
英文摘要
Symmetric powers of matroids were introduced by Lovász and Mason. The subject has recently attracted renewed interest through its connection with the realizability of tropical linear spaces by tropical ideals. Anderson proved that the tropical linear space associated with a matroid $M$ is the variety of a tropical ideal if and only if $M$ has a $d$th symmetric power for every positive integer $d$. He also asked whether the Vámos matroid $V_8$ has a second symmetric power. We answer this question in the negative by proving that $V_8$ has no second symmetric power. By Anderson's characterization, the associated tropical linear space cannot be realized as the variety of a tropical ideal.
