TOPICS
Search

Jacobian Ideal

The Jacobian ideal J_f of a polynomial or formal power series f is the ideal generated by its partial derivatives:

J_f=((partialf)/(partialx_1),...,(partialf)/(partialx_n)).

For f in k[[x_1,...,x_n]] with all partial derivatives vanishing at the origin, that critical point is isolated iff the quotient ring k[[x_1,...,x_n]]/J_f is finite-dimensional over k. The Euler conductor is J_f:f, and the Tjurina ideal is (J_f,f) (Zhang and Zuo 2026).

See also

Euler Conductor, Ideal, Tjurina Ideal

Explore with Wolfram|Alpha

References

Zhang, Y. and Zuo, H. "The Hassanzadeh-Nasrollah Nejad-Simis Conjecture on Euler Conductors." 7 Oct 2026. https://arxiv.org/abs/2610.09833.

Cite this as:

Weisstein, Eric W. "Jacobian Ideal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacobianIdeal.html

Subject classifications