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.
Consider a linear realization of a matroid over a field. One associates with it a configuration
polynomial and a symmetric bilinear form with linear homogeneous coefficients.
The corresponding configuration hypersurface and its non-smooth locus support the
respective first and second degeneracy scheme of the bilinear form.We showthat these
schemes are reduced and describe the effect of matroid connectivity: for (2-)connected
matroids, the configuration hypersurface is integral, and the second degeneracy scheme
is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also
the second degeneracy scheme is integral. In the process, we describe the behavior
of configuration polynomials, forms and schemes with respect to various matroid
constructions.
Liegruppen
(1997)