Betti numbers of random hypersurface arrangements
Betti numbers of random hypersurface arrangements
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随机超曲面排列的贝蒂数
DOI：
10.1112/jlms.12658
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Natarajan, Abhiram
中科院分区：
文献类型：
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作者：
Basu, Saugata;Lerario, Antonio;Natarajan, Abhiram
我们研究了RPn$\mathbb {R}\mathrm{P}^n$中随机多项式（根据Kostlan分布分布）的零点排列的Betti数的预期行为。使用随机谱序列，我们证明了RPn$\mathbb {R}\mathrm{P}^n$中s$s$此类超曲面的补集中连通分量的预期数量的渐进精确估计。我们也调查了同样的问题的情况下，超曲面的定义由随机二次多项式。在这种情况下，我们建立了一个连接的贝蒂数的这种安排与预期的行为的某个模型的随机定义的几何图形。虽然我们的一般结果意味着，平均第零贝蒂数的工会随机超曲面安排是有界的，从上面的一个函数，线性增长的多项式的数量的安排，使用与随机图的连接，我们显示了一个上界的预期第零贝蒂数的随机二次安排，是次线性的多项式的数量的安排。这个界限是一个普遍的结果，在我们的随机图模型，这可能是独立的利益的预期数量的连接组件。
We study the expected behavior of the Betti numbers of arrangements of the zeros of random (distributed according to the Kostlan distribution) polynomials in RPn$\mathbb {R}\mathrm{P}^n$. Using a random spectral sequence, we prove an asymptotically exact estimate on the expected number of connected components in the complement of s$s$ such hypersurfaces in RPn$\mathbb {R}\mathrm{P}^n$. We also investigate the same problem in the case where the hypersurfaces are defined by random quadratic polynomials. In this case, we establish a connection between the Betti numbers of such arrangements with the expected behavior of a certain model of a randomly defined geometric graph. While our general result implies that the average zeroth Betti number of the union of random hypersurface arrangements is bounded from above by a function that grows linearly in the number of polynomials in the arrangement, using the connection with random graphs, we show an upper bound on the expected zeroth Betti number of random quadrics arrangements that is sublinear in the number of polynomials in the arrangement. This bound is a consequence of a general result on the expected number of connected components in our random graph model which could be of independent interest.