= Solution
Orient both <spheres> in the standard way and normalize their <area forms> to total area $4\pi$. The <degree of a map between oriented manifolds> can be obtained in two ways. For a <regular value> $y$, use the <degree as a sum of local degrees>:
$$
\boxed{\deg g=\sum_{x\in g^{-1}(y)}\operatorname{sign}\det Dg_x.}
$$
Each inverse image is isolated by the <inverse function theorem>, and compactness makes the set finite. The determinant is computed in consistently oriented local coordinates. A second method is <spherical degree by area pullback>:
$$
\boxed{\deg g=\frac1{4\pi}\int_{S^2}g^*\omega
=\frac1{4\pi}\int_0^{2\pi}\!\int_0^\pi
g\cdot(\partial_\theta g\times\partial_\varphi g)\,d\theta\,d\varphi,}
$$
where the last formula represents $g$ as a unit vector in $\mathbb R^3$. The <pullback of a differential form> already contains the signed <Jacobian determinant>; no extra $\sin\theta$ is to be inserted in that last coordinate expression.
To relate the methods, replace $\omega$ by a smooth top-degree <differential form> with the same total integral supported in a small neighbourhood of a <regular value>. Two such top-degree forms with equal integral differ by an <exact differential form> on $S^2$, by its top-degree <de Rham cohomology>. Their pullbacks therefore have the same integral by <Stokes theorem>. Over the chosen neighbourhood, $g$ splits into local inverse branches; the <change of variables formula> makes the contribution of each branch its <orientation> sign times $4\pi$. Their sum is precisely the first formula. Thus the area integral is an integer and agrees with the signed inverse-image count.
For a nonconstant <rational map>, first use <common-factor reduction of a rational map> so $p$ and $q$ are coprime. Write $k=\max(\deg p,\deg q)$ for these reduced polynomials. A generic finite target value $w$ has inverse images at the roots of $p-wq$: avoiding exceptional values makes its degree $k$ and its roots simple. The <fundamental theorem of algebra> supplies $k$ roots. A <holomorphic map> has positive real <Jacobian determinant> $|R'|^2$ at a regular point, so every local sign is $+1$. \b[Hence]
$$
\boxed{\deg R=k\quad\text{for a coprime representation}.}
$$
The source leaves coprimality implicit. In an unreduced representation the answer is $\max(\deg p,\deg q)-\deg\gcd(p,q)$, including degree zero for a constant reduced map. For example $(z^2-1)/(z-1)$ extends to $z+1$ and has degree one, although the unreduced maximum degree is two. Exceptional inverse images at infinity or multiple roots do not change the <degree of a rational map of the Riemann sphere>.
For the <rational map approximation for Skyrmions>, use <stereographic projection> $z=\tan(\theta/2)e^{i\varphi}$ and the unit target vector
$$
\mathbf n_R=\frac{(2\operatorname{Re}R,\,2\operatorname{Im}R,\,1-|R|^2)}{1+|R|^2}.
$$
Combine this <rational map> with a radial profile to form a <special unitary group> field:
$$
U(r,z)=\cos f(r)\,\mathbf1+i\sin f(r)\,\mathbf n_R(z)\cdot\boldsymbol\sigma,
\qquad f(0)=\pi,\quad f(\infty)=0,
$$
where $\boldsymbol\sigma$ are the <Pauli matrices>. The endpoint values make $U(0)=-\mathbf1$ independent of angle and $U(\infty)=\mathbf1$. Appropriate radial behaviour gives an admissible <finite-energy field configuration>. With $L_i=U^\dagger\partial_iU$, choose the <topological baryon number in the Skyrme model> convention
$$
B=-\frac1{24\pi^2}\int\epsilon_{ijk}\operatorname{tr}(L_iL_jL_k)\,d^3x.
$$
Separating the radial and angular factors gives
$$
\boxed{B=-\frac{2k}{\pi}\int_0^\infty f'(r)\sin^2f(r)\,dr=k.}
$$
Thus the <degree of a rational map of the Riemann sphere> supplies the <Skyrmion> charge.
In conventional dimensionless massless <Skyrme model> units, its static energy reduces to
$$
E=4\pi\int_0^\infty\left[r^2f'^2+2k(1+f'^2)\sin^2f+\mathcal I[R]\frac{\sin^4f}{r^2}\right]dr,
\qquad\mathcal I[R]=\frac1{4\pi}\int_{S^2}J_R^2\,d\Omega,
$$
with the <angular Jacobian of a rational map>
$$
J_R=\left[\frac{1+|z|^2}{1+|R|^2}|R'|\right]^2,
\qquad\frac1{4\pi}\int J_R\,d\Omega=k.
$$
The <Cauchy-Schwarz inequality> gives $\mathcal I\geq k^2$. These formulas follow from the radial strain $|f'|$ and the two equal angular strains $\sin f\sqrt{J_R}/r$: the quadratic energy sums their squares and the quartic <Skyrme term> sums their pairwise products of squares. Minimize the <angular integral in the rational map approximation> over degree-$k$ maps, then minimize the remaining radial energy with the stated endpoints. This replaces a three-dimensional field minimization by finitely many map coefficients and an <ordinary differential equation> for $f$.
\b[The method constructs a charge-$k$ variational approximation], with topology built in and with <rotational symmetry of a rational map> translated into combined spatial and <isospin rotations>. It is efficient for identifying shapes and providing initial data for unrestricted numerical relaxation. Its restrictions are equally concrete: it uses one radial profile and a holomorphic angular map independent of radius, so it cannot represent arbitrary radial-angular correlations, separated clusters, or all deformations. Apart from the degree-one <Skyrmion hedgehog ansatz>, it generally does not solve the full field equation exactly. Massive-pion terms can be included in the radial functional but do not remove these restrictions, and multi-shell or unrestricted fields may be needed for larger charges. Approximate energy minima and a final <collective-coordinate quantization> are distinct steps.
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