On minimal free resolutions and the method of shifted partial derivatives in complexity theory

The minimal free resolution of the Jacobian ideals of the determinant polynomial were computed by Lascoux [12], and it is an active area of research to understand the Jacobian ideals of the permanent, see e.g., [13, 9]. As a step in this direction we compute several new cases and completely determine the linear strands of the minimal free resolutions of the ideals generated by sub-permanents. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory, building on the use of Hilbert functions in [8]. We also compute several Hilbert functions relevant for complexity theory.