发表机构
KU Leuven（荷语鲁汶大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广拟阵可提升性至可实现截断情形，引入泛可实现性，通过Oxley--Wang导出拟阵统一刻画，并应用于铺砌拟阵及场景分析，给出新证明。
AI 中文摘要
可提升性最初是针对铺砌拟阵引入的，用于系统地构造相关拟阵簇理想中的多项式。本文将可提升性的概念推广到具有可实现截断的拟阵。我们还引入了拟阵的泛可实现性概念；这是一种强可实现性，它蕴含拟阵的可实现性和可提升性。我们研究了在具有固定截断的拟阵格中，弱序下可提升性和泛可实现性的行为。我们通过泛拟阵实现的Oxley--Wang导出拟阵，发展了一个研究拟阵可提升性的统一框架。我们用Oxley--Wang导出拟阵来刻画可提升性和泛可实现性。此外，我们给出了泛Oxley--Wang导出拟阵的基的代数描述，该描述以拟阵实现空间的坐标截面表示，并将我们的构造与拟阵的松弛实现空间联系起来。作为我们框架的一个应用，我们获得了可提升且泛可实现的铺砌拟阵的组合刻画，并将其与Whiteley的场景分析理论联系起来。特别地，我们的方法通过Oxley--Wang导出拟阵，给出了Whiteley关于允许非平凡（尖锐）场景的超图刻画的新的证明。最后，我们计算了一类特殊的稀疏铺砌拟阵的泛Oxley--Wang导出拟阵。
英文摘要
Liftability was first introduced for paving matroids, where it was used to systematically construct polynomials in the ideals of the associated matroid varieties. In this paper, we generalise the notion of liftability to matroids with realisable truncations. We additionally introduce the notion of generic realisability of a matroid; a strong notion of realisability that implies both realisability and liftability of the matroid. We study the behaviour of liftability and generic realisability under the weak order of matroids within the lattice of matroids with fixed truncation. We develop a unified framework for studying liftability of matroids through Oxley--Wang derived matroid of generic matroid realisations. We characterise liftability and generic realisability in terms of Oxley--Wang derived matroids. Moreover, we give an algebraic description of the bases of the generic Oxley--Wang derived matroids in terms of coordinate sections of matroid realisation spaces and relate our constructions to slack realisation spaces of matroids. As an application of our framework, we obtain a combinatorial characterisation of liftable and generically realisable paving matroids and connect it with Whiteley's theory of scene analysis. In particular, our approach gives a new proof, via Oxley--Wang derived matroids, of Whiteley's characterisation of hypergraphs admitting non-trivial (sharp) scenes. Finally, we compute the generic Oxley--Wang derived matroid for a special class of sparse paving matroids.
Comments36 pages, 7 figures
