Liegruppen
(1997)
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.
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.