On the existence of continuous (approximate) roots of algebraic equations |
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Authors: | Kazuhiro Kawamura Takeshi Miura |
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Institution: | a Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan b Department of Basic Technology, Applied Mathematics and Physics, Yamagata University, Yonezawa 992-8510, Japan |
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Abstract: | The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm ∥⋅∥∞. The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.Also we consider continuous approximate roots of the equation zn−f=0 with respect to z, where f∈C(X), and provide a topological characterization of compact Hausdorff space X with dimX?1 such that the above equation has an approximate root in C(X) for each f∈C(X), in terms of the first ?ech cohomology of X. |
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Keywords: | 46J10 54F65 |
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