Rational functions with poles bounded at one point

Existence of closed points and the residue degree. By [F1] the curve $C$ is nonempty with a unique generic point ${\eta }_C$, and its chain dimension one provides a strict chain of two nonempty irreducible closed subsets, so $C$ has at least two points and we may fix a point $p_0\mathrm{\ne }{\eta }_C$; by [F1] every point other than ${\eta }_C$ is closed, so $p_0$ is a closed point of $C$. In particular closed points exist, and for every closed point $x$, such as the given $p$, the residue field is finite over $k$ with $[\kappa (x):k]\ge 1$ by [F2]; the given residue degree is $d=[\kappa (p):k]\ge 1$.

The dimension bound. By [F4] applied to the divisor $np$, whose degree is ${\mathrm{deg}}_k(np)=nd$ by [F2], one has $l(np)\ge nd+1-g\ge 2$; by [F3] the integer $l(np)$ is the $k$-dimension of $L(np)$, so $L(np)$ is a $k$-vector space of dimension at least two.

A nonconstant function with poles only at $p$. By [F5] the constants form the subspace $k\cdot 1\subseteq L(np)$ of dimension one; since ${\dim }_{k}L(np)\ge 2>1$, [F6] provides $f\in L(np)$ with $f\mathrm{\notin }k\cdot 1$, so $f$ is not a constant function. By [F3] membership $f\in L(np)$ means $\mathrm{div}(f)+np\ge 0$, so ${\mathrm{ord}}_x(f)\ge 0$ for every closed point $x\mathrm{\ne }p$ and ${\mathrm{ord}}_p(f)\ge -n$; that is, every pole of $f$ lies at $p$ with order at most $n$.

$f$ has a pole, and the pole divisor. Since $f$ is nonconstant by step 2.1, $f\mathrm{\notin }L(0)$: otherwise $\mathrm{div}(f)\ge 0$ and [F5] would exhibit $f$ as a constant. Hence some order ${\mathrm{ord}}_x(f)$ is negative; by step 2.1 the only point where this can happen is $p$, so $f$ has a pole at $p$. Therefore the pole divisor $(f{)}_{\mathrm{\infty }}={\sum }_{{\mathrm{ord}}_x(f)<0}(-{\mathrm{ord}}_x(f))[x]$ is a nonzero effective divisor supported at $p$, with coefficient $-{\mathrm{ord}}_p(f)$ between $1$ and $n$.

Conclusion and choice accounting. For the given closed point $p$ and every $n\ge 1$ with $nd+1-g\ge 2$, steps 2.1 and 3.1 produce $f\in L(np)$ that is nonconstant, has all its poles at $p$ of order at most $n$, and has at least one pole there; moreover every $f\in L(np)\setminus k\cdot 1$ has the same properties by steps 2.1 and 3.1 applied to it. The Axiom of Choice is inherited from the suppliers recorded in [F8], including the divisor and dimension interfaces [F2], [F3], [F6], [F7]; choosing the single function $f$ outside the one-dimensional subspace $k\cdot 1$ is a single selection from a nonempty set and needs no choice principle, and the flagged dictionary [F7] records the inherited obligation on ${\mathcal{O}}_C(D)$.