Let k be a field of characteristic 0, let n≥ 1, and let f∈ (x_1,…,x_n)^2⊂ k[[x_1,…,x_n]] define an isolated critical point at the origin. If J_f=(partial f/partial x_1,…,partial f/partial x_n) is the Jacobian ideal, then J_f:f not⊆ (J_f,f); equivalently, there exists g∈ k[[x_1,…,x_n]] such that gf∈ J_f but g notin (J_f,f).

Hassanzadeh–Nasrollah Nejad–Simis conjecture on Euler conductors

The conjecture of Hassanzadeh, Nasrollah Nejad, and Simis asserts, in characteristic zero, that for an isolated hypersurface singularity with Jacobian ideal $J_f$ , one has $J_f:f\not\subseteq (J_f,f)$ . The preprint reformulates this through annihilators and Jordan blocks.

Known results

Hassanzadeh, Nasrollah Nejad, and Simis established the plane-curve case; the higher-dimensional statement was left as a conjecture.

Unrefereed preprint (date not stated)

Yizhi Zhang and Huaiqing Zuo claim the characteristic-zero conjecture, and also claim the stronger local noncontainment $J_f:f\not\subseteq (J_f,f)+\mathfrak n(J_f:f)$ . They give an isolated singularity over $\mathbb{Q}[[x_1,\ldots,x_5]]$ for which $\overline{J_f}=\mathfrak m^3$ and $J_f:f\subseteq\overline{J_f}$ , refuting the proposed integral-closure strengthening.

Current status (as of October 2026): The characteristic-zero conjecture is claimed proved, but remains unverified; the proposed integral-closure strengthening is claimed false by an explicit example.

Hassanzadeh–Nasrollah Nejad–Simis conjecture on Euler conductors

Primary source

Additional references

Progress summary

Solutions 0