Yano's Conjecture for Two-Puiseux-Pair Irreducible Plane Curve Singularities

Abstract

In 1982, Tamaki Yano proposed a conjecture predicting the $b$-exponents of an irreducible plane curve singularity germ that is generic in its equisingularity class. In this article, we prove the conjecture for the case in which the irreducible germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. This 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.