多项式根与开映射
Polynomial Roots and Open Mappings
摘要：
讨论了复多项式映射的“开性”，并将其应用于代数基本定理。在这一类下降证明S。德军总部Thompson，J. Milnor，and S. Reich-S。斯梅尔这些证明考虑了多项式的临界点。对D. Reem和F.S.卡特使FTA的一个很短的证明成为可能，而无需参考导数的零点。我们把高斯的Helmstedt论文（1799）看作是应用微分拓扑的一个练习，并填写了S. Gersten和J. Stallings。多项式函数可以被扰动或重新对齐（Martin、Savitt和Singer的工作）以消除临界值，因此我们使用实平面代数曲线的配置。本文对隐函数定理的处理是利用收敛幂级数的Banach代数上的压缩算子。Walter.除了泛函分析之外，关键的几何洞察力是拓扑横截性结果，该结果导致严格证明平面曲线相交为高斯和A。奥斯特洛夫斯基说他们会的。在解析映射的范畴中工作使得“隐式”解的唯一性更加透明，并且曲线的多项式（和调和）性质对极值的数量施加了有限性，这来自于Be'zout定理的一个众所周知的应用。我们认为这提供了一个澄清高斯的声明，一个代数曲线进入一个域，必须再次离开，这是更具体的比宣布这样的曲线是一个“一维流形”。
英文摘要：
The "openness" of a complex polynomial mapping is discussed and applied to the Fundamental Theorem of Algebra. In this category fall proofs of S. Wolfenstein, R.L. Thompson, J. Milnor, and S. Reich-S. Smale. These proofs take into account the critical points of the polynomial. New elementary proofs of openness due to D. Reem and F.S. Cater make possible a very short proof of FTA without reference to zeros of the derivative. We regard Gauss's Helmstedt Thesis (1799) as an exercise in applied differential topology, and fill out a synopsis published by S. Gersten and J. Stallings. The polynomial function may be perturbed or re-aligned (work of Martin, Savitt and Singer) to eliminate critical values so we work with a configuration of real, plane algebraic curves. The treatment we give to the Implicit Function Theorem uses contractive operators on the Banach algebra of convergent power series according to W. Walter. In addition to Functional Analysis, the key geometric insight is a topological transversality result that leads to a rigorous demonstration that the plane curves intersect as Gauss and A. Ostrowski said they would. Working in the category of analytic mappings makes uniqueness of "implicit" solutions more transparent, and the polynomial (and harmonic) nature of the curves imposes finiteness on the number of extrema, coming from a well-known application of the Be'zout Theorem. We think this provides a clarification of Gauss's statement that an algebraic curve entering a domain, must leave again, that is more concrete than declaring such a curve to be a "one-dimensional manifold".
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