Weierstrass varieties and coverings over compact groups

We consider Weierstrass polynomials over Banach algebras of continuous complex-valued functions on compact groups. Weierstrass varieties are defined by roots of such polynomials. The talk is devoted to results that reveal an intimate relation between Weierstrass varieties and finite-sheeted coverings over compact abelian groups. We also discuss applications of these results to studying the structure of coverings over compact groups.