arXiv 2108.063772021-08math.CO

前进的道路：极值组合学中的热带化

A Path Forward: Tropicalization in Extremal Combinatorics

Grigoriy Blekherman, Annie Raymond

备注：: 45 pages, 4 figures, corrected the proof of Theorem 2.22 and improved the proof of Lemma 2.2

摘要：

极值组合学中的许多重要问题可以表述为证明图同态数中的纯二项式不等式，即，证明了hom $(H_1,G)^{a_1}\cdots$ hom $(H_k,G)^{a_k}\geq$ hom $(H_{k+1},G)^{a_{k+1}}\cdots$ hom $(H_m,G)^{a_m}$对某些固定图$H_1,\dots,H_m$和所有图$G$成立.一个突出的例子是Sidorenko的猜想。对于固定的图集$\mathcal{U}=\{H_1,\dots,H_m\}$，$\mathcal{U}$的图中的有效纯二项不等式的指数向量形成凸锥.我们计算这个锥的几个家庭的图形，包括完整的图形，甚至周期，明星和道路，后者是最有趣和复杂的情况下，我们计算。在所有这些情况下，我们观察到一个诱人的多面体现象：有效的纯二项式不等式的锥实际上是有理多面体，因此所有有效的纯二项式不等式可以从极端射线的指数向量的有限集合中生成。利用Kopparty和Rossman的工作，我们表明，锥的有效不等式确实是合理的多面体时，所有的图形$H_i$是串并联和弦，我们猜想，多面体持有任何有限的集合$\mathcal{U}$。我们证明了多面体现象也发生在拟阵和单纯复形中。我们的描述的不平等的道路涉及到一个推广的Erdens-Simonovits猜想最近证明在其原来的形式Saintelam和一个新的家庭的不平等没有观察到以前。我们还解决了Kopparty和Rossman关于路的同态控制指数的一个公开问题。我们的主要工具之一是热带化，这是复代数几何中的一种众所周知的技术。我们证明了几个结果的热带化，这可能是独立的利益。

英文摘要：

Many important problems in extremal combinatorics can be be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that hom$(H_1,G)^{a_1}\cdots$hom$(H_k,G)^{a_k}\geq$hom$(H_{k+1},G)^{a_{k+1}}\cdots$hom$(H_m,G)^{a_m}$ holds for some fixed graphs $H_1,\dots,H_m$ and all graphs $G$. One prominent example is Sidorenko's conjecture. For a fixed collection of graphs $\mathcal{U}=\{H_1,\dots,H_m\}$, the exponent vectors of valid pure binomial inequalities in graphs of $\mathcal{U}$ form a convex cone. We compute this cone for several families of graphs including complete graphs, even cycles, stars and paths; the latter is the most interesting and intricate case that we compute. In all of these cases, we observe a tantalizing polyhedrality phenomenon: the cone of valid pure binomial inequalities is actually rational polyhedral, and therefore all valid pure binomial inequalities can be generated from the finite collection of exponent vectors of the extreme rays. Using the work of Kopparty and Rossman, we show that the cone of valid inequalities is indeed rational polyhedral when all graphs $H_i$ are series-parallel and chordal, and we conjecture that polyhedrality holds for any finite collection $\mathcal{U}$. We demonstrate that the polyhedrality phenomenon also occurs in matroids and simplicial complexes. Our description of the inequalities for paths involves a generalization of the Erdős-Simonovits conjecture recently proved in its original form by Sağlam and a new family of inequalities not observed previously. We also solve an open problem of Kopparty and Rossman on the homomorphism domination exponent of paths. One of our main tools is tropicalization, a well-known technique in complex algebraic geometry. We prove several results about tropicalizations which may be of independent interest.

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