广义剖分与Monsky定理
Generalized Dissections and Monsky's Theorem
摘要：
Monsky的著名等剖定理如下从他更普遍的证明存在的多项式关系$f$之间的地区的三角形的解剖单位平方米。最近，作者研究了一个不同的多项式$p$，也是在这样的解剖三角形的面积之间的关系，这是在一定的解剖变形不变。本文研究了这两个多项式之间的关系。我们首先推广了剖分的概念，允许方向与平面方向不同的三角形。我们定义了这些广义剖分的变形空间，我们表明，这个空间是一个不可约的代数簇。然后，我们将Monsky定理推广到广义解剖的背景下，表明Monsky多项式$f$可以被选择为变形下不变的。虽然$f$不是唯一定义的，但$p$和$f$之间的相互作用允许我们识别多项式$f$的一对规范选择。在许多情况下，规范$f$多项式的所有系数都是正的。我们还使用变形不变性的$f$证明了多项式$p$是同余模2的一个权力的总和，其变量。
英文摘要：
Monsky's celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial $p$, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monsky's polynomial $f$ can be chosen to be invariant under deformation. Although $f$ is not uniquely defined, the interplay between $p$ and $f$ then allows us to identify a canonical pair of choices for the polynomial $f$. In many cases, all of the coefficients of the canonical $f$ polynomials are positive. We also use the deformation-invariance of $f$ to prove that the polynomial $p$ is congruent modulo 2 to a power of the sum of its variables.
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