Linear relations on the coefficients of the linking polynomial

In 1972, Brylawski showed that the coefficients of the Tutte polynomial of a matroid are not independent, but they obey some linear relations. This result was extended to matroid perspectives by Vergnas in 1999. We extend this result further to all matroids pairs, and we conjecture that all the linear relations obeyed by the coefficients of the linking polynomial are linear combinations of the “basic” ones.