Compatibility of annihilators under contraction vs. deletion

Investigate whether, for any matroid M and element e ∈ E(M) that is not a coloop, the annihilator ideals of the basis generating polynomials satisfy Ann(f_{M \ e}) ⊆ Ann(f_{M / e}), i.e., whether operators annihilating the deletion polynomial also annihilate the contraction polynomial.

Background

The author explores whether operators that annihilate the deletion polynomial f_{M\e} necessarily annihilate the contraction polynomial f_{M/e}. An affirmative answer would facilitate constructing homomorphisms between Gorenstein rings A(M\e) and A(M) and support decomposition strategies.