Weierstrass polynomials and coverings of compact groups (Q350845)

Let \(G\) be a connected compact abelian group. A Weierstrass polynomial of degree \(n\in\mathbb{N}\) over \(G\) is a mapping \(R:G\times \mathbb{C}\rightarrow \mathbb{C}\) of the form \[ R(g,z)=z^n+\sum_{j=1}^nf_j(g)z^{n-j}, \] where \(g\in G\), \(z\in\mathbb{C}\) and \(f_j\) are continious functions \(G\rightarrow \mathbb{C}\). Such polynomial defines a covering \(X\rightarrow G\), where \[ X=\{(g,z)\in G\times \mathbb{C}\mid R(g,z)=0\}, \] and the mapping is induced by the projection \[ (g,z)\mapsto g. \] The main result of the paper is the following theorem. It is proved that each finite-sheeted covering of the group \(G\) by a conneted space \(X\) is equivalent to a covering defined by finitely many Weierstrass polynomials of the form \(z^n-\chi\), where \(\chi\) is a character of \(G\).