On the existence of continuous (approximate) roots of algebraic equations

Abstract. The present paper considers the existence of continu-ous 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 con-tinuous 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 alge-braically 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 Čech cohomology of X. 1. Introduction, Result