标题： 理想转换系统 显示英文标题

标题： Ideal transition systems

摘要： 这篇半教学性的文章介绍了一种归纳方法，用于计算多项式环中理想族的初始理想和Gröbner基。 该方法从给定的一组对$(I,J)$开始，其中$I$是任意理想，$J$是包含在$I$的初始理想中的单项式理想。 如果能够在所选理想之间建立特定的过渡递归关系，这些包含关系就会变成一个等式系统。 我们描述了在两个有启发性的案例中此类系统的显式构造——即矩阵Schubert流形的理想及其反对称类似物。 尽管与这些例子有许多形式上的相似之处，但对于矩阵Schubert流形的对称版本，构建相同类型的过渡系统仍是一个开放问题。 我们提出了几个若能构建此类系统便可得出的猜想，同时讨论了在对称情况下出现的特殊障碍。 显示英文摘要

摘要： This semi-expository article presents an inductive method of computing initial ideals and Gr\"obner bases for families of ideals in a polynomial ring. This method starts from a given set of pairs $(I,J)$ where $I$ is any ideal and $J$ is a monomial ideal contained in the initial ideal of $I$. These containments become a system of equalities if one can establish a particular transition recurrence among the chosen ideals. We describe explicit constructions of such systems in two motivating cases -- namely, for the ideals of matrix Schubert varieties and their skew-symmetric analogues. Despite many formal similarities with these examples, for the symmetric versions of matrix Schubert varieties, it is an open problem to construct the same kind of transition system. We present several conjectures that would follow from such a construction, while also discussing the special obstructions arising in the symmetric case.