Braid monodromy for a trinomial algebraic equation by means of Mellin-Barnes integral representations

TL;DR

This paper develops a braid monodromy framework for solutions of trinomial algebraic equations using Mellin-Barnes integrals, providing insights into their Galois groups and braid structures.

Contribution

It introduces a novel approach to analyze trinomial algebraic equations through braid monodromy and Mellin-Barnes integrals, connecting algebraic solutions with braid group representations.

Findings

Braid monodromy representations are constructed for trinomial equations.

The monodromy around branch points is described via rational twists of strands.

The Galois group of the trinomial algebraic equation is explicitly characterized.

Abstract

We establish a braid monodromy representation of functions satisfying an algebraic equation containing three terms (trinomial equation). We follow global analytic continuation of the roots to a trinomial algebraic equation that are expressed by Mellin-Barnes integral representations. We depict braids that arise from the monodromy around all branching points. The global braid monodromy is described in terms of rational twists of strands that yield a classical Artin braid representation of algebraic functions. As a corollary, we get a precise description of the Galois group of a trinomial algebraic equation.