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Euler Conductor

The Euler conductor of a formal power series f in R=k[[x_1,...,x_n]] is the ideal quotient

J_f:f={g in R:gf in J_f},

where J_f=(partialf/partialx_1,...,partialf/partialx_n) is the Jacobian ideal. The associated Tjurina ideal is I_f=(J_f,f).

Zhang and Zuo (2026) proved the Hassanzadeh-Nasrollah Nejad-Simis conjecture that, if k has field characteristic zero and f in (x_1,...,x_n)^2 has an isolated critical point, then

J_f:f !subset= (J_f,f).

Here the isolated-critical-point condition means that R/J_f is finite-dimensional over k. Zhang and Zuo (2026) acknowledge ChatGPT 6 Astra assistance with proof strategy and calculations and state that they verified the argument.

See also

Formal Power Series, Ideal Quotient, Jacobian Ideal, Tjurina Ideal

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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. "Euler Conductor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulerConductor.html

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