关于具有偏序集参数化的单有理簇
On unirational varieties with poset parameterizations
AI总结：
研究利用偏序集为特定有理映射下像的扎里斯基闭包提供参数化，将代数几何等领域中多个问题转化为组合学问题，通过各领域实例展示该技术的应用。
AI中文摘要：
我们使用偏序集（posets）为半代数集在坐标函数为非负整数系数多项式的有理映射下的像的扎里斯基闭包提供一种规范参数化。这种单有理簇的偏序集参数化使我们能够将几个经过充分研究的问题转化为组合学问题，比如将问题简化为描述与该簇相关的偏序集。这些问题包括代数几何中的隐式化问题、环面重新参数化问题、簇的线性跨度计算以及区分同一环境空间中两个半代数子集的问题。该技术适用于这些问题在多个领域的实例，包括代数几何、代数组合学、统计学和应用代数。我们通过来自每个领域的例子展示了该技术，包括割线簇的退化子簇、拟阵平坦簇（它推广了边多面体的环面簇）以及多变量数据分析和进化生物学中出现的簇。
英文摘要：
We use partially ordered sets (posets) to provide a canonical parameterization for the Zariski closure of the image of a semialgebraic set under a rational map whose coordinate functions are polynomials with nonnegative integral coefficients. The resulting poset parametrization of such a unirational variety allows us to translate several well-studied problems into combinatorics; e.g. reducing the problems to describing the poset associated to the variety. These problems include, the implicitization problem from algebraic geometry, the toric reparameterization problem, the computation of the linear span of the variety, and the problem of distinguishing two semialgebraic subsets of the same ambient space. The technique applies to instances of these problems in several fields, including algebraic geometry, algebraic combinatorics, statistics and applied algebra. We demonstrate the technique on examples from each field, including degenerate subvarieties of secant varieties, matroid flat varieties -- which generalize toric varieties of edge polytopes, as well as varieties arising in multivariate data analysis and evolutionary biology.
