Suppose you have an affine noetherian scheme $X=SpecA$. We are given:
- Two global sections $f,g \in A$ such that $f(p)=g(p)=0$ in $k(p)$ for all $p \in X$.
- Two real constants $C,D \in \mathbb{R}\setminus \mathbb{Q}$.
For every $p \in X, s \in A$, define $v_p(s) \in \mathbb{N}$ such that $s_p \in (pA_p)^{v_p(s)} \setminus (pA_p)^{v_p(s)+1}$, and $\infty$ if $s=0$.Note that in a local noetherian ring $\bigcap_{n} (pA_p)^n = 0$, so we covered all possible cases.
Define the set $$ S(f,g,C,D) = \{p \in X: v_p(f), v_p(g) < \infty \ \text{and } C \le \frac{v_p(f)}{v_p(g)} \le D \} $$
Note that $C,D \in \mathbb{R} \setminus \mathbb{Q}$ so that you can put $C \le$ or $C<$ as you like. We can divide by $v_p(g)$ because $v_p(g) > 0$ (i.e., $g(p) =0$) for every $p \in X$. My question is:
Has this set nice topological properties? It is locally closed or something like that? Can something be said about its dimension when ranging $C,D$?
Explicitly, in my problem I want to determine $g$. I am assigned an increasing succession $C_n \in \mathbb{R}$ and a prime number $q \in \mathbb{Z}$. Given a prime $p \in X$, suppose $n$ is (the only number) such that $p \in S(q,g,C_n,C_{n+1})$ (the $p$'s such that $v_p(q)= \infty$ or $v_p(g)= \infty$ are easy to treat, so I exclude them). Then the more $n$ is high, the more is difficult for me to determine $g_p$ (i have something like "higher-order" terms which appear). I was wondering how to constrain the range of this $n$ with some hypothesis on the ring, other than asking for boundedness of $v_p(q)$ when ranging over $p \in X$ ($q$ is fixed).
Thank you and I am sorry if my language is awkward. I am new to the scheme world. Any reference to the study of this kind of valuations (in not normal context) are welcomed.