Combinatorics of Sphere Covers and the Shift-Incidence Matrix
نویسنده
چکیده
Most problems in algebra, at some stage, must consider algebraic equations. Sometimes solving them, other times producing them – so their existence manifests a desired structure. Akin to doing differential equations, a researcher in algebraic equations must specialize to get somewhere. My speciality is algebraic equations in two variables, say w and z, where one variable, say z, interprets as a data variable. In this proposal I take relations with coefficients in the complex numbers C. Such an algebraic equation belongs to a unique equivalence class of compact covers of the Riemann sphere, Pz = Cz ∪{∞} (with Cz the standard z-plane of complex variables). The word cover here means a (ramified) analytic (nonconstant) map from a compact Riemann surface to the sphere. The notation for such a cover is φ : X → Pz. Points {z1, . . . , zr} over which the cover ramifies are called branch points.
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