Linear differential operators for generic algebraic curves

TL;DR

This paper presents an efficient method to construct linear differential operators with polynomial coefficients for algebraic curves, enabling the analysis of all branches of functions without solving the algebraic equation, even with non-solvable Galois groups.

Contribution

It introduces a novel computational approach to derive differential operators for algebraic curves without solving the defining equations, applicable to complex Galois groups.

Findings

Method efficiently constructs differential operators

Applicable to algebraic curves with non-solvable Galois groups

Enables analysis of all branches of algebraic functions

Abstract

We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic curve. The proposed method does not require solving the algebraic equation and can be applied in the case when its Galois group is not solvable.