Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-08-28
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The degree of a rational self-map of the Riemann sphere

Definition

Let R:C^→C^ be rational. Choose coprime polynomials P,Q∈C[z], not both zero, with R(z)=P(z)/Q(z) on the finite chart. The degree of R is deg⁡R:=max⁡(deg⁡P,deg⁡Q). For a finite constant map this gives degree 0.

Multiplying P and Q by the same nonzero scalar does not change the maximum, so the degree is well defined on the rational map rather than on a chosen representative pair.

Depends on

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Sources