The Monodromy problem for hyperelliptic curves

TL;DR

This paper investigates the monodromy problem for hyperelliptic curves, focusing on polynomials of the form y^4+g(x), and extends results to direct sums of degree four polynomials, revealing conditions for monodromy to generate full homology.

Contribution

It generalizes previous work on hyperelliptic curves by solving the monodromy problem for specific polynomial forms and their direct sums, providing new insights into monodromy generation.

Findings

Solved the monodromy problem for y^4+g(x) polynomials.

Extended results to direct sums of degree four polynomials.

Identified conditions under which monodromy generates the entire homology.

Abstract

We study the Dynkin diagram associated to the monodromy of direct sums of polynomials. The monodromy problem asks under which conditions on a polynomial, the monodromy of a vanishing cycle generates the whole homology of a regular fiber. We consider the case $y^4+g(x)$, which is a generalization of the results of Christopher and Marde$\check{s}$i\'c about the monodromy problem for hyperelliptic curves. Moreover, We solve the monodromy problem for direct sums of fourth degree polynomials.