Two Properties of Higher‐Order Nash Blow‐Up Local Algebras

In this paper, we study the higher‐order Nash blow‐up local algebras associated with isolated hypersurface singularities, particularly focusing on the nonexistence of negative weighted derivations. Building on our earlier conjecture, which was proved by Badilla‐Céspedes, Castorena, Duarte, and Núñez‐Betancourt, we propose a new conjecture that generalizes our previous one. Inspired by the result of Badilla‐Céspedes et al., we further demonstrate that the results in this paper validate our new conjecture partially. Besides, from the perspective of minors, we elucidate the structure of higher‐order Nash blow‐up algebras, which constitutes another main result of this paper. These findings contribute to the broader understanding of singularities and extend the applicability of higher‐order Nash blow‐up local algebras.