Representing tropical linear spaces by circuits

三维空间中的热带椭圆曲线

Tropical elliptic curves in 3-space

本研究通过四维Cayley多胞体的405246030个正则单位模三角剖分，分类得到热带三维空间中两二次曲面相交形成的4009个16顶点16边三价图，对应椭圆曲线的极大退化稳定模型。

Minimal tropical basis for Bergman fan of matroid

The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and $R_{10}$. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.

关于具有唯一极小热带基的简单拟阵

On simple matroids with a unique minimal tropical basis

该研究刻画了简单拟阵的热带基，证明闭回路必含于所有热带基，进而给出具有唯一极小热带基的简单拟阵的特征，解决了Matroid Union Blog的相关问题。

带标量乘法门的热带电路

Tropical Circuits with Scalar Multiplication Gates

研究带标量乘法门的热带电路，证明其计算最大权重有向生成树和最大权重二分完美匹配时的指数规模下界，得出单调和非单调最大输出神经网络的指数规模分离，揭示有强制凸性约束的神经网络模型有时需指数级更大才能表达相同函数。

拟阵在域的热带扩张之上

Matroids over tropical extensions of tracts

本文研究域的热带扩张上的拟阵，推广了赋值拟阵与初始拟阵的对应关系，证明了旗拟阵和正拟阵的类似结论，并利用富化赋值证明了富化热带线性空间的结构定理。

Infinite graphic matroids Part I

An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graph-like topological space: that is, a graph-like space in the sense of Thomassen and Vella. This extends Tutte's characterization of finite graphic matroids. The representation we construct has many pleasant topological properties. Working in the representing space, we prove that any circuit in a 3-connected graphic matroid is countable.