  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.
