标题： 多边形三角剖分空间和蒙斯奇多项式 显示英文标题

标题： Spaces of polygonal triangulations and Monsky polynomials

摘要： 给定一个$n$-边形的组合三角剖分，我们研究（a）所有可能的平面绘制，其中边是直线段且边界具有固定形状，以及（b）此类绘制中三角形面积的可能性代数流形。 我们定义了一个广义的三角剖分概念，并证明了在正方形的广义三角剖分$\T$中，三角形的面积必须满足一个仅依赖于$\T$组合的不可约齐次多项式关系$p(\T)$。 不变量$p(\T)$被称为\emph{蒙斯凯多项式}；它捕捉了关于$\T$的代数、几何和组合信息。 我们给出一个计算$p(\T)$的次数下界的算法，并且我们给出了几个使用该算法计算次数的例子。 显示英文摘要

摘要： Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for the areas of the triangles in such drawings. We define a generalized notion of triangulation, and we show that the areas of the triangles in a generalized triangulation $\T$ of a square must satisfy a single irreducible homogeneous polynomial relation $p(\T)$ depending only on the combinatorics of $\T$. The invariant $p(\T)$ is called the \emph{Monsky polynomial}; it captures algebraic, geometric, and combinatorial information about $\T$. We give an algorithm that computes a lower bound on the degree of $p(\T)$, and we present several examples in which the algorithm is used to compute the degree.