Higher Nash blowup on normal toric varieties

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non-singular points. In the case of normal toric varieties we give a combinatorial description of the higher Nash blowup in terms of a Gr\"obner fan. This description will allow us to prove the analogue of Nobile's theorem on the usual Nash blowup in this context. More precisely, we prove that for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular.