论文标题
广义解剖和蒙斯基定理
Generalized Dissections and Monsky's Theorem
论文作者
论文摘要
蒙斯基著名的等距定理是从他更普遍的证据证明了三角形区域中多项式关系$ f $的存在。最近，作者研究了不同的多项式$ p $，这也是在这种解剖中的三角形区域之间的关系，在解剖的某些变形下是不变的。在本文中，我们研究了这两个多项式之间的关系。 我们首先概括了解剖的概念，允许其方向与平面不同的三角形。我们定义了这些广义解剖的变形空间，我们表明该空间是不可还原的代数品种。然后，我们将Monsky的定理扩展到广义解剖的上下文，表明蒙斯基的多项式$ F $可以选择在变形下不变。尽管$ f $不是唯一的定义，但是$ p $和$ f $之间的相互作用使我们能够确定多项式$ f $的规范对选择。在许多情况下，规范$ f $多项式的所有系数都是正面的。我们还使用$ f $的变形不变性来证明多项式$ p $是其变量总和的幂。
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.
