Higher Jacobian ideals, contact equivalence and motivic zeta functions

We show basic properties of higher Jacobian matrices and higher Jacobian ideals for functions and apply it to obtain two results concerning singularities of functions. Firstly, we prove that a higher Nash blowup algebra is invariant under contact equivalences, which was recently conjectured by Hussain, Ma, Yau and Zuo. Secondly, we obtain an analogue of a result on motivic nearby cycles by Bussi, Joyce and Meinhardt.