Common-factor reduction of a rational map ($\mathrm{deg}R=\max (\mathrm{deg}p,\mathrm{deg}q)-\mathrm{deg}\gcd (p,q)$)

Cancel the common polynomial factors of $p$ and $q$ before computing the degree of a rational map of the Riemann sphere. Removable common zeros do not contribute sheets. The reduced maximum polynomial degree is the number of inverse images of a generic target value, counted with positive holomorphic local degrees.