I have a question that I hope has a clear answer. I will begin by introducing some definitions and an example to help readers follow, assuming some background knowledge based on the tags I'm using.
In the paper Algebroid plane curves whose Milnor and Tjurina numbers differ by one or two, the author defines the local Milnor number and the local Tjurina number in terms of the ring of power series $\mathbb{K}[[x,y]]$.
Then, the paper considers a curve $f \in \mathbb{K}[[x]][y]$ having one place at infinity—that is, its meromorphic series is irreducible in $\mathbb{K}((x))[y]$. To this curve, we associate a semigroup $\mathcal{S}$, and suppose the parametrization of the curve is given by: $$ x(t) = t^n, \quad y(t) = t^m + c_\lambda t^\lambda + \cdots $$ where $\lambda$ is the Zariski invariant.
We define the algebra $$ \mathcal{A} = \mathbb{K}[x(t), y(t)], $$ and the module $$ \mathcal{M} = \mathcal{A} \cdot x'(t) + \mathcal{A} \cdot y'(t). $$
Now define: $$ d(\mathcal{A}) = \{ \deg(f) \mid f \in \mathcal{A} \setminus \{0\} \}, \quad d(\mathcal{M}) = \{ \deg(f) \mid f \in \mathcal{M} \setminus \{0\} \}, \quad \mathcal{S} = d(\mathcal{A}). $$
We say that $m \in d(\mathcal{M})$ is a non-exact element if $m+1 \notin \mathcal{S}$. Let $\operatorname{ne}(\mathcal{M})$ denote the number of non-exact elements. The differentials corresponding to these degrees are called non-exact differentials. It can be shown that the difference between the local Milnor number and the local Tjurina number associated to $f$ is equal to $\operatorname{ne}(\mathcal{M})$.
Now, my question is the following.
On page 67 of the above-mentioned paper, the author states:
For these curves, a nonexact differential with minimal value $\lambda + n - 1$ is given by $m y\, dx - n x\, dy$.
A straightforward computation shows that this differential has degree $\lambda + n - 1$ and is non-exact.
But how do we know that $\lambda + n - 1$ is the minimal non-exact element in $d(\mathcal{M})$?
A second, related question: If we consider a polynomial curve instead of a local (power series) curve, can we find the minimal non-exact element?
For instance, in the paper Canonical bases of modules over one dimensional K-algebras, the authors consider the global case and mention the same differential on page 1135. However, they work with it without necessarily assuming it is the minimal one.
For instance, let $\mathcal{S} = d(\mathcal{A}) = \langle 3, 4 \rangle$, so the set of gaps (i.e., $\mathbb{N} \setminus \mathcal{S}$) is $\{1, 2, 5\}$, and the corresponding gap degrees for possible non-exact elements are $\{0, 1, 4\}$. Suppose $\lambda = 2$, and consider the polynomial curve defined by the parametrization: $$ x(t) = t^4 + p_2 t^2 + p_1 t, \quad y(t) = t^3 + q_1 t. $$
How can one explicitly determine the minimal non-exact element associated with this polynomial curve? That is, how do we find the smallest $m \in d(\mathcal{M})$ such that $m + 1 \notin \mathcal{S}$?
