Exact degree of a polynomial curve
Definition
Let \(X(t) = t^n + \alpha_1t^{n-1} + \cdots + \alpha_n, Y(t) = t^m + \beta_1 t^{m-1} + \cdots + \beta_m\) be two polynomials of \(\mathbb{K}[t]\) and \(\mathbf{A} = \mathbb{K}[X(t), Y(t)]\). Let \(f(X,Y)\) be the unique irreducible polynomial of \(\mathbb{K}[X, Y]\), monic in \(Y\), such that \(f(X(T), Y(T)) = 0\). If \(\mathbf{M} = X'(t)\mathbf{A} + Y'(t)\mathbf{A}\), it is defined the set
\[ I = \{ord_t(F) ~ | ~ F \in \mathbf{M}\}. \]
It can be proven that \(I\) is a relative ideal of \(S = \Gamma(f)\), where \(\Gamma(f)\) is the numerical semigroup associated to f. It is said that \(i \in I\) is an exact degree if \(i+1 \in S\). The other elements are said non exact degrees of \(\mathbf{M}\). The set of non exact degrees is denoted by \(NE(\mathbf{M})\).