= Solution
An <irreducible complex analytic hypersurface> in a <complex manifold> $X$ is a closed irreducible analytic subset of pure complex codimension one. A <local defining function of a complex analytic hypersurface> $Y$ at $x$ is a holomorphic function $f$ on a neighbourhood $U$ such that
$$
Y\cap U=\{f=0\}.
$$
The necessary local algebra is that the stalk $\mathcal O_{X,x}$ is a <regular local ring>, hence a <unique factorization domain>, and that the local branches of a hypersurface germ determine finitely many height-one <prime ideals>. Each is principal; the product of their generators gives $f$, and removing repeated factors makes it reduced. This also covers a globally irreducible hypersurface that has several local branches at a singular point.
A <divisor on a complex manifold> is a locally finite formal sum $D=\sum_Yn_YY$ of irreducible analytic hypersurfaces with integer coefficients. On a sufficiently small $U_\alpha$, local defining functions give a meromorphic equation $f_\alpha$ for $D$. The quotients $f_\alpha/f_\beta$ are nowhere-zero holomorphic functions. Gluing frames by
$$
e_\alpha=(f_\beta/f_\alpha)e_\beta
$$
produces the <holomorphic line bundle associated to a divisor> $[D]$, and $f_\alpha e_\alpha$ gives its canonical meromorphic section with divisor $D$.
The <Euler sequence on complex projective space>
$$
0\longrightarrow\mathcal O
\longrightarrow\mathcal O(1)^{\oplus(n+1)}
\longrightarrow T\mathbb{CP}^n
\longrightarrow0
$$
implies $\det T\mathbb{CP}^n\cong\mathcal O(n+1)$. Taking the dual determinant yields the <canonical bundle of complex projective space>
$$
K_{\mathbb{CP}^n}\cong\mathcal O(-n-1)\cong[-(n+1)H],
$$
where $H$ is a <hyperplane divisor>.
The hypotheses on the homogeneous polynomial $p$ say that
$$
V=\{p=0\}\subset\mathbb{CP}^n
$$
is a smooth <projective hypersurface> of degree $k$, so its divisor line bundle is $[V]\cong\mathcal O(kH)$. The <Adjunction formula> gives
$$
K_V
\cong\bigl(K_{\mathbb{CP}^n}\otimes[V]\bigr)|_V
\cong\mathcal O_V(k-n-1)
\cong[(k-n-1)H|_V].
$$
This is the <canonical bundle of a smooth projective hypersurface>.
Now fix an isomorphism $\Phi:[P]\to[Q]$ and regard $s_P$ and $\Phi^{-1}s_Q$ as <holomorphic section>[holomorphic sections] of the same <holomorphic line bundle>. They have no common zero because $P\ne Q$. Their homogeneous coordinates therefore define a well-defined <holomorphic map>
$$
F:S\longrightarrow\mathbb{CP}^1,
\qquad
F(x)=[s_P(x):\Phi^{-1}s_Q(x)].
$$
In a local frame, the quotient
$$
f=\frac{s_P}{\Phi^{-1}s_Q}
$$
is a <meromorphic function> with divisor $(f)=P-Q$. Thus $F^{-1}(0)=P$ and $F^{-1}(\infty)=Q$, both with multiplicity one. The <degree of a holomorphic map> $F$ is therefore one. A nonconstant degree-one holomorphic map between compact connected <Riemann surfaces> is a <biholomorphism>, so the displayed map is biholomorphic.
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