Column expansion identities and quadratic spanning forest identities

Column expansion identities of determinants give a source of quadratic spanning forest polynomial identities and allow us determine the dimension of the space of certain quadratic spanning forest identities, settling a conjecture of one of us with Vlasev from 2012. Furthermore, we give a combinatorial interpretation of such spanning forest identities via an edge-swapping argument previously developed by one of us in 2019. Quadratic spanning forest polynomials identities are of particular interest because they are useful for quantum field theory calculations in four dimensions.