AI 中文总结
2024年Cisto等人引入多集合和多集合理想。本文将曲折游走等概念从胞腔集合扩展到多集合，定义自然多集合。关注其多集合理想有二次格罗贝纳基的情况，利用零和条件证明素性，还表明坐标环的h多项式等与切换车多项式等相等，回答了相关猜想和问题。
AI 中文摘要
2024年，Cisto等人引入了多集合和多集合理想，分别推广了胞腔集合和内部2-子式理想的概念。本文自然地将关于曲折游走、零和条件、车数和切换多项式的概念和结果从胞腔集合扩展到多集合，并定义了一类称为自然多集合的多集合，其结构类似于胞腔集合。我们关注其多集合理想关于由顶点上某些特定顺序诱导的字典序允许二次格罗贝纳基的自然多集合。利用零和条件，我们证明了这类多集合的素性，这表明Mascia等人提出的曲折猜想对于满足上述性质的胞腔集合成立。此外，我们表明这类多集合的坐标环的h多项式和正则性分别等于它们的切换车多项式和车数。这些结果对Jahangir和Navarra提出的切换车多项式猜想以及Rinaldo和Romeo提出的车数开放问题给出了肯定答案。
英文摘要
In 2024, Cisto et al. introduced polyocollections and polyocollection ideals, which generalize the notions of collections of cells and inner $2$-minors ideals, respectively. In this paper, we naturally extend the notions and results about zig-zag walk, zero-sum condition, rook number, and switching polynomial from collections of cells to polyocollections, and we define a class of polyocollections, called natural polyocollections, whose structure is similar to that of collections of cells. We focus on natural polyocollections whose polyocollection ideals admit quadratic Gröbner bases with respect to lexicographic order induced by some specific orders on vertices. Using the zero-sum condition, we prove the primality of this class of polyocollections, which shows that the zig-zag conjecture proposed by Mascia et al. holds true for collections of cells satisfying the aforementioned properties. Moreover, we show that $h$-polynomials and regularities of the coordinate rings of this class of polyocollections are equal to their switching rook polynomials and rook numbers, respectively. These results give positive answers to the switching rook polynomial conjecture proposed by Jahangir and Navarra, and the rook number open problem proposed by Rinaldo and Romeo for collections of cells satisfying the aforementioned properties.
