Yano's conjecture for 2-Puiseux pair irreducible plane curve singularities

TL;DR

This paper proves Yano's conjecture for a specific class of irreducible plane curve singularities with two Puiseux pairs and distinct monodromy eigenvalues, linking $b$-exponents to Bernstein polynomial roots.

Contribution

It confirms Yano's conjecture in the case of two Puiseux pairs with a particular monodromy condition, providing explicit Bernstein polynomial roots.

Findings

Yano's conjecture holds for two Puiseux pairs with distinct monodromy eigenvalues.

The $b$-exponents match the negatives of Bernstein polynomial roots.

Explicit computation of Bernstein polynomial roots for the case.

Abstract

In 1982, Yano proposed a conjecture predicting the $b$-exponents of an irreducible plane curve singularity which is generic in its equisingularity class. In this article we prove the conjecture for the case of two Puiseux pairs and monodromy with distinct eigenvalues. The hypothesis on the monodromy implies that the $b$-exponents coincide with the opposite of the roots of the Bernstein polynomial, and we compute the roots of the Bernstein polynomial.