Linear differential operators for polynomial equations
Let \(k_0\) be a number field and \(\overline{k_0}\) be its algebraic closure. Let \(P\in k_0(x)[y]\) be a squarefree polynomial in \(y\). The derivation \(\delta=\frac d{dx}\) extends uniquely to the algebraic closure \(\overline{k_0(x)}\) of \(k_0(x)\). The minimal operator associated with \(P\) is defined as the monic differential operator \(L_P=\delta^t+a_{t-1}\delta^{t-1}+\dots+a_0\) with \(a_i\in k_0(x)\) of smallest positive order such that \(L_P(y)=0\) for all roots of \(P\) in \(\overline{k_0(x)}\). Some algorithms for calculating this operator are given. This allows one to calculate the genus of the complex curve defined by \(P=0\) (when \(P\) is absolutely irreducible), the absolute factorization of \(P\) over \(\overline{k_0}\), and calculate information concerning the Galois group of \(P\) over \(\overline{k_0}(x)\) as well as over \(k_0(x)\).
scientific article; zbMATH DE number 1886757
scientific article