Newton-Puiseux exponents of an irreducible polynomial

Definition

The definition of Newton-Puiseux exponents comes from the following result.

Proposition

Let \(f(x,y) = y^n + a_1(x)y^{n-1} + \cdots + a_n(x) \in \mathbb{K}((x))[y]\). Suppose that \(f(x,y)\) is irreducible. There exists \(y(t) \in \mathbb{K}((t))\) such that \(f(t^n, y(t)) = 0\). Furthermore,

  1. \(f(t^n, y) = \prod_{w, w^n = 1} (y - y(wt))\),

  2. if \(w_1 \ne w_2\) and \(w_1^n = w_2^n = 1\), then \(y(w_1t) \ne y(w_2t)\),

  3. \(gcd(n, gcd(Supp(y(t)))) = 1\),

where \(Supp(y(t))\) is the support of \(y(t)\).

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 \(m_1 = \min \{p \in Supp(y(t)) ~ | ~ d_1 \not | \ \ p\}\), \(d_2 = gcd(d_1, m_1)\). Then for all \(i \ge 2\), if \(d_i \ne 1\), let \(m_i = \min \{p \in Supp(y(t)) ~ | ~ d_i \not | \ \ p \}\) and \(d_{i+1} = gcd(d_i, m_i)\). By the above proposition, \(m_i\) is well defined and there exists \(h \ge 1\) such that \(d_{h+1} = 1\). It is defined the Newton-Poiseux exponents of \(f\) as \((m_1, m_2, \ldots, m_h)\) and is denoted by \(\underline{m}\).

References

Assi, Abdallah, Marco D’Anna, and Pedro A. Garcı́a-Sánchez. 2020. Numerical Semigroups and Applications. Vol. 3. RSME Springer Series. Springer, [Cham]. https://doi.org/10.1007/978-3-030-54943-5.