Linear Differential Operators for Polynomial Equations

The results of this paper spring from the elementary fact that an algebraic function satisfies a linear differential equation. Let k0 be a number field and k0 be its algebraic closure. Let P ∈ k0(x)[y] be a squarefree polynomial of degree n in y. The derivation δ = d dx extends uniquely to the algebraic closure k0(x) of k0(x). We define the minimal operator associated with P to be the monic differential operator LP = δ + at−1δ + · · · + a0 with ai ∈ k0(x) of smallest positive order such that LP (y) = 0 for all roots of P in k0(x). In Section 2, we give algorithms to calculate this operator. In Section 3, we assume that P is absolutely irreducible, that is, irreducible over k0(x). We show that information derived from the singular points of the minimal operator allows one to give a simple formula (and direct method) to calculate the genus of P = 0. In Section 4 we give two methods to factor a polynomial P ∈ k0(x)[y] over k0(x). Together with the algorithm in Section 3, this yields a new polynomial time algorithm for this task. In Section 5, we discuss how the minimal operator allows us to find properties of the Galois groups of P over k0(x) and over k0(x). In the appendix we