Let $R$ be a noetherian commutative ring of dimension one which is reduced. Moreover, $R$ is free of rank $n$ over the polynomial ring $k[x]$ where $k$ is a field.
The background: $R = \mathcal{O}_X(U)$ is the affine coordinate ring where $U \subseteq X$ is an affine everywhere dense open subset of $X$, a reduced projective curve over $k$ with finite surjective morphism onto $\mathbb{P}^1_k$.
Fix a basis $y_1,\ldots,y_n$ of $R$ over $k[x]$. For any $y \in R$ let $\deg(y)$ be the maximal degree of its coefficients regarding the fixed basis of $R$. Let $P_1,\ldots,P_r$ be the minimal primes of $R$ such that $\cup_i P_i \neq R$. Thus there are elements of $R$ not being zero divisors.
What can we say about the upper bound on the minimum of the degrees of regular elements of $R$? What is $$\min_{y \in R}\ \{ \deg(y) \mid y \notin \cup_i P_i \}?$$
I am looking for some bounds in terms of $n$ and $g$, where $g$ is the arithmetic genus of $X$. Bounds in terms of the genera of the irreducible components belonging to the $P_i$ are also welcome.
Does anyone have some helpful references or ideas how to approach this kind of problem?
Remark: Originally, the question arised in terms of elements of fractional ideals $I$ of $R$, which are also free modules over $k[x]$ and do not only consist of zero divisors. Hence the question above is the case $I = R$.