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

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 z n − f = 0 with respect to z, where f ∈ C ( X ) , and provide a topological characterization of compact Hausdorff space X with dim X ⩽ 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.