Chapter IV - ALGEBRA AND TOPOLOGY OF WEIERSTRASS POLYNOMIALS

Summary

Throughout this chapter, X denotes a connected and locally pathwise connected topological space with the homotopy type of a CW–complex. Let C(X) denote the ring of complex valued, continuous functions on X. A Weierstrass polynomial P(x, z) over X can then be viewed as an element in the polynomial ring C(X)[z].

A natural first question concerning the algebra of a Weierstrass polynomial P(x, z) over X is to ask whether it has a root over C(X), in other words, whether there exists a continuous function λ : X → ℂ such that P(x, λ(x)) = 0. In §1, we shall present the basic elements of work of Gorin and Lin containing necessary and sufficient conditions that a space X has to satisfy in order that every simple Weierstrass polynomial P(x, z) of degree n ≥ 2 over X splits completely into linear factors over C(X). Associated with a simple Weierstrass polynomial P(x, z) over X we have the polynomial covering map T. E → X. Complete solvability of the equation P(x, z) = 0 is equivalent to triviality of π: E → X. This is an example of the connections between the algebra of the simple Weierstrass polynomial P(x, z) on the one hand and the topology of the polynomial covering map x: E → X on the other hand.

For an arbitrary covering map π. E → X, there is an induced monomorphism of rings π*: C(X) → C(E). Thereby we can consider C(E) as a ring extension of C(X), or as a C(X)–algebra.