Math 2.0

Algebraic geometry Affine and computational geometry Conjecture

Jacobian conjecture

Builds on Characteristic of a ringPolynomial ringJacobian matrixJacobian determinant

Statement

Let kk be a field of characteristic 00 and let n≥1n \ge 1. Let F1,…,Fn∈k[x1,…,xn]F_1, \ldots, F_n \in k[x_1, \ldots, x_n] be polynomials, and let F=(F1,…,Fn) ⁣:kn→knF = (F_1, \ldots, F_n) \colon k^n \to k^n. Suppose that the Jacobian determinant det⁡(∂Fi∂xj)\det \left( \frac{\partial F_i}{\partial x_j} \right) is a nonzero constant in kk. Then there are polynomials G1,…,Gn∈k[x1,…,xn]G_1, \ldots, G_n \in k[x_1, \ldots, x_n] such that G=(G1,…,Gn)G = (G_1, \ldots, G_n) satisfies G∘F=F∘G=id⁡G \circ F = F \circ G = \id.

Remarks

The converse holds: if FF has a polynomial inverse GG, the chain rule gives det⁡JF⋅(det⁡JG∘F)=1\det J_F \cdot (\det J_G \circ F) = 1, so det⁡JF\det J_F is a unit of the polynomial ring, that is, a nonzero constant. The conjecture is open for every n≥2n \ge 2; the case n=1n = 1 is easy, since a polynomial with constant nonzero derivative has degree 11. The question was posed by Ott-Heinrich Keller in 1939. Characteristic 00 is needed: over 𝔽p\F_p, the polynomial F(x)=x−xpF(x) = x - x^p has derivative 11, but it has degree pp and is not invertible by a polynomial. It is known that the conjecture holds when all FiF_i have degree at most 22. Bass, Connell and Wright showed that it suffices to prove it, for all nn, for maps of the form F=x+HF = x + H with HH homogeneous of degree 33.

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  1. Characteristic of a ring concept
  2. Euclidean space object
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  6. Function of several variables concept
  7. Partial derivative concept
  8. Jacobian matrix concept
  9. Jacobian determinant concept
  10. Jacobian conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Jacobian_conjecture
About this page
TagB5JFXEQQD8
URIm2://B5JFXEQQD8
Kindconjecture
WritingW0 stub
MSC14R15, 13F20, 14R10
StatusThis copy of the site has no build data. Run m2 status B5JFXEQQD8.
AboutJacobian determinant, Polynomial ring
UsesCharacteristic of a ring, Polynomial ring, Jacobian matrix, Jacobian determinant
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Changed2 November 2026

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