Hypersurface singularities with monomial Jacobian ideal

TL;DR

This paper classifies certain hypersurface singularities with monomial Jacobian ideals, showing they are equivalent to Thom-Sebastiani polynomials, and explores their combinatorial and analytic properties.

Contribution

It proves that power series with monomial extended Jacobian ideals are right equivalent to Thom-Sebastiani polynomials, solving a problem by Hauser and Schicho.

Findings

Power series with monomial extended Jacobian ideals are right equivalent to Thom-Sebastiani polynomials.

Introduces a Jacobian semigroup ideal involving a transversal matroid.

Power series with quasihomogeneous extended Jacobian ideals are strongly Euler homogeneous.

Abstract

We show that every convergent power series with monomial extended Jacobian ideal is right equivalent to a Thom-Sebastiani polynomial. This solves a problem posed by Hauser and Schicho. On the combinatorial side, we introduce a notion of Jacobian semigroup ideal involving a transversal matroid. For any such ideal we construct a defining Thom-Sebastiani polynomial. On the analytic side, we show that power series with a quasihomogeneous extended Jacobian ideal are strongly Euler homogeneous. Due to a Mather-Yau-type theorem, such power series are determined by their Jacobian ideal up to right equivalence.