On minimal free resolutions and the method of shifted partial derivatives in complexity theory Academic Article uri icon

abstract

  • The minimal free resolution of the Jacobian ideals of the determinant polynomial were computed by Lascoux [22], and it is an active area of research to understand the Jacobian ideals of the permanent, see eg,[23, 18]. 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 an exploration of the utility and limits of the method of shifted partial derivatives introduced in [17, 13]. The method of shifted partial derivatives amounts to computing Hilbert functions of Jacobian ideals, and the Hilbert functions are in turn the Euler characteristics of the minimal free resolutions of the Jacobian ideals. We compute several such Hilbert functions relevant for complexity theory. We show that the method of shifted partial derivatives alone cannot prove the padded permanent …

publication date

  • April 1, 2015