有效化的代数基本定理：通过Sturm链的初等实代数证明
The Fundamental Theorem of Algebra made effective: an elementary real-algebraic proof via Sturm chains
AI总结：
本文利用Sturm链和Cauchy指标，在实闭域上给出了代数基本定理的初等实代数证明，并构造了代数求根算法。
摘要：
Sturm定理（1829/35）提供了一个优雅的算法来计数和定位任何实多项式的实根。在残数演算（1831/37）中，Cauchy将Sturm方法扩展到计数和定位任何复多项式的复根。对于全纯函数，Cauchy指标基于围道积分，但在多项式的特殊情况下，可以通过Sturm链使用欧几里得除法像实情况一样有效计算。通过这种方式，我们为任意实闭域上的多项式提供了Cauchy定理的代数证明。作为主要工具，我们在实代数环境中形式化了Gauss的几何绕数概念（1799），由此推导出代数基本定理的实代数证明。该证明是初等的，因为它仅使用介值定理和实多项式的算术。因此，它可以用实闭域的一阶语言表述。此外，该证明是构造性的，并立即转化为代数求根算法。
英文摘要：
Sturm's theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any real polynomial. In his residue calculus (1831/37) Cauchy extended Sturm's method to count and locate the complex roots of any complex polynomial. For holomorphic functions Cauchy's index is based on contour integration, but in the special case of polynomials it can effectively be calculated via Sturm chains using euclidean division as in the real case. In this way we provide an algebraic proof of Cauchy's theorem for polynomials over any real closed field. As our main tool, we formalize Gauss' geometric notion of winding number (1799) in the real-algebraic setting, from which we derive a real-algebraic proof of the Fundamental Theorem of Algebra. The proof is elementary inasmuch as it uses only the intermediate value theorem and arithmetic of real polynomials. It can thus be formulated in the first-order language of real closed fields. Moreover, the proof is constructive and immediately translates to an algebraic root-finding algorithm.
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