Matroids over a domain admit a Grothendieck-Tutte polynomial with the deletion-contraction property, and the Hilbert series of the associated face module specializes that polynomial.
Poincar\'e polynomial of elliptic arrangements is not a specialization of the Tutte polynomial
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abstract
The Poincar\'e polynomial of the complement of an arrangements in a non compact group is a specialization of the $G$-Tutte polynomial associated with the arrangement. In this article we show two unimodular elliptic arrangements (built up from two graphs) with the same Tutte polynomial, having different Betti numbers.
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Set of independencies and Tutte polynomial of matroids over a domain
Matroids over a domain admit a Grothendieck-Tutte polynomial with the deletion-contraction property, and the Hilbert series of the associated face module specializes that polynomial.