论文标题
随机单体理想的渐近程度
Asymptotic Degree of Random Monomial Ideals
论文作者
论文摘要
连接代数和几何形状的基本不变性之一是理想的程度。在本文中，我们针对在\ cite {rmi}中定义的随机单型理想的多功能Erdős-rényi-type模型得出了程度的概率行为。我们研究了与单一理想相关的楼梯结构，并表明在随机情况下，楼梯图的形状大致相当夸张，并且在几种随机模型中，这种行为是可靠的。由于该楼梯下的离散体积与数字理论中研究的省级高阶分隔函数有关，因此我们使用这种联系和对楼梯图的形状的控制来得出随机单体理想的渐近程度。计算单一理想程度的另一种方法是标准对分解。本文在环变量的任何子集的随机单元理想的标准对数量上得出边界。由最大子集索引的标准对给出了程度的计数，并且是随机单体理想的更细微的不变。
One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In this paper we derive the probabilistic behavior of degree with respect to the versatile Erdős-Rényi-type model for random monomial ideals defined in \cite{rmi}. We study the staircase structure associated to a monomial ideal, and show that in the random case the shape of the staircase diagram is approximately hyperbolic, and this behavior is robust across several random models. Since the discrete volume under this staircase is related to the summatory higher-order divisor function studied in number theory, we use this connection and our control over the shape of the staircase diagram to derive the asymptotic degree of a random monomial ideal. Another way to compute the degree of a monomial ideal is with a standard pair decomposition. This paper derives bounds on the number of standard pairs of a random monomial ideal indexed by any subset of the ring variables. The standard pairs indexed by maximal subsets give a count of degree, as well as being a more nuanced invariant of the random monomial ideal.
