# Analytic continuation for solutions to the system of trinomial algebraic equations

## TL;DR

This paper develops methods for analytically continuing solutions to trinomial algebraic systems using Mellin-Barnes integrals and Puiseux expansions, providing explicit continuation formulas.

## Contribution

It introduces a novel approach combining Mellin-Barnes integrals and polyhomogeneity to extend solutions of trinomial systems analytically.

## Key findings

- Constructed Puiseux expansions for analytic continuation
- Demonstrated the use of Mellin-Barnes integrals in extension
- Extended solutions to the universal trinomial system

## Abstract

In the paper, we deal with the problem of getting analytic continuations for the monomial function associated with a solution to the reduced trinomial algebraic system. In particular, we develop the idea of applying the Mellin-Barnes integral representation of the monomial function for solving the extension problem and demonstrate how to achieve the same result following the fact that the solution to the universal trinomial system is polyhomogeneous. As a main result, we construct Puiseux expansions (centred at the origin) representing analytic continuations of the monomial function.