Higher-Order Nash Blowups and the Resolution of Singularities in Arbitrary Dimension

In their groundbreaking work, Castillo, Duarte, Leyton-Álvarez, and Liendo demonstratethat the classical (order-1) Nash blowup fails to resolve singularities in dimensions four andhigher, over fields of arbitrary characteristic. We show that this obstruction is intrinsic tothe first-order nature of the construction. By enriching the Nash blowup with higher-orderdifferential data—specifically, using the theory of Hasse–Schmidt derivations in positivecharacteristic and higher-order jets in characteristic zero—we break the closed orbits thatcause the failure. We provide a formal proof that the second-order Nash blowup resolvesthe characteristic-zero counterexample, and that the Hasse–Schmidt Nash blowup of orderp resolves the counterexamples in characteristic p. This suggests that a canonical resolution algorithm based on Nash’s original vision is viable when equipped with sufficientdifferential information.