Order of a formal series
Definition
Let \(\mathbb{K}\) be a field, \(x\) a variable and let \(\mathbb{K}[[x]]\) be the set of formal series on \(x\) with coefficients in \(\mathbb{K}\), that is, the set of elements of the form \(\sum_{i \in \mathbb{N}} a_i x^i\) with \(a_i \in \mathbb{K}\) for all \(i \in \mathbb{N}\).
Let \(f(x) = \sum_{i \in \mathbb{N}} a_i x^i \in \mathbb{K}[[x]]\). It is defined the order of \(f\) as the smallest \(i \in \mathbb{N}\) such that \(a_i \ne 0\), and it is denoted by \(ord_x(f(x))\). For \(f(x) = 0\), it is usually written \(ord_x(0) = + \infty\).
Examples
\(\circ\) Let \(f(x) = 7x^5 + 4x^3 - x^2\). The coefficients are: \(a_0 = a_1 = 0, a_2 = -1, a_3 = 4, a_4 = 0\) and \(a_5 = 7\). The order of \(f(x)\) is \(2\) since \(a_0 = a_1 = 0\) and \(a_2 = -1 \ne 0\).
\(\circ\) Let \(S\) be a numerical semigroup and let us consider \(f(x) = \sum_{s \in S \setminus \{0\}} x^s\). The order of \(f\) is the smallest non-zero coefficient, equivalent to compute the minimum of \(S\setminus \{0\}\), which is the multiplicity of \(S\), \(m(S)\). Therefore, \(ord_x(f) = m(S)\).