Characteristic sequences associated to an irreducible polynomial
The definition of characteristic sequences comes from the following result.
Let us consider \(f(x,y) \in \mathbb{K}((x))[y]\) irreducible and let \(y = \sum_{p \in \mathbb{N}} c_p t^p \in \mathbb{K}((t))\).
Let \(d_1 = n = deg_y(f)\) and let \(m_1 = \min \{p \in Supp(y(t)) ~ | ~ d_1 \not \mid \ \ p\}\), \(d_2 = gcd(d_1, m_1)\). Then for all \(i \ge 2\), if \(d_i \ne 1\), let \(m_i = \min \{p \in Supp(y(t)) ~ | ~ d_i \not \mid \ \ p \}\) and \(d_{i+1} = gcd(d_i, m_i)\). By the above preposition, \(m_i\) is well defined and there exists \(h \ge 1\) such that \(d_{h+1} = 1\). It is defined \(\underline{m} = (m_1, m_2, \ldots, m_h)\) and \(\underline{d} = (d_1, d_2, \ldots, d_{h+1})\).
It is also defined \(e_i = \frac{d_i}{d_{i+1}}\) for all \(i \in \{1,2, \ldots, h\}\).
Finally, it is defined the sequence \(\underline{r} = (r_0, r_1, \ldots, r_h)\) as follows: \(r_0 = n, r_1 = m_1\) and for all \(i \in \{2, 3, \ldots, h\}\),
\[ r_i = r_{i-1}e_{i-1} + m_i - m_{i-1}. \]
It is defined the characteristic sequences of \(f\) as the sequences \(\underline{m}, \underline{d}\) and \(\underline{r}\). In particular, the sequence \(\underline{m}\) is called the Newton-Puiseux exponents of \(f\).