论文标题
通过迭代完全解决五重奏
Completely solving the quintic by iteration
论文作者
论文摘要
在十九世纪后期，费利克斯·克莱因（Felix Klein）从加洛伊斯（Galois）放入的垂死状态下解决了解决五重奏方程的问题。克莱因（Klein）的方法是代数和几何形状的混合物，建立在常规二十面体的结构上。他的方法的关键特征是五重奏的Galois组与二十面体的旋转对称性之间的联系。 克莱因（Klein）作品大约一个世纪以来，P。Doyle和C. McMullen开发了一种用于解决五重奏的算法，也利用了二十面体对称性。他们的创新是在一个复杂变量中采用对称的动力系统。实际上，动态行为为多项式对称性的部分断裂和在MAP的一次迭代运行之后提取了两个根。 最近发现的地图，该地图的动力破坏了所有Quintic的对称性，从而使五个根部从一次运行中浮出水面。在勾勒出一些代数和几何背景之后，讨论可以从完全意义上提出一个明确的程序来解决五重奏。
In the late nineteenth century, Felix Klein revived the problem of solving the quintic equation from the moribund state into which Galois had placed it. Klein's approach was a mix of algebra and geometry built on the structure of the regular icosahedron. His method's key feature is the connection between the quintic's Galois group and the rotational symmetries of the icosahedron. Roughly a century after Klein's work, P. Doyle and C. McMullen developed an algorithm for solving the quintic that also exploited icosahedral symmetry. Their innovation was to employ a symmetrical dynamical system in one complex variable. In effect, the dynamical behavior provides for a partial breaking of the polynomial's symmetry and the extraction of two roots following one iterative run of the map. The recent discovery of a map whose dynamics breaks all of the quintic's symmetry allows for five roots to emerge from a single run. After sketching some algebraic and geometric background, the discussion works out an explicit procedure for solving the quintic in a complete sense.
