Jacobian conjecture

In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller. It was later named and widely publicised by Shreeram Abhyankar, as an example of a question in the area of algebraic geometry that requires little beyond a knowledge of calculus to state.

Related Topics:
Mathematics - Polynomial - Variable - Ott-Heinrich Keller - Shreeram Abhyankar - Algebraic geometry - Calculus

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For fixed N > 1 consider N polynomials Fi, for 1 ≤ i ≤ N in the variables

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:X1, ?, XN,

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and with coefficients in the complex numbers C. The Jacobian determinant J of the Fi, considered as a vector-valued function

Related Topics:
Coefficient - Complex number - Jacobian determinant - Function

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:F: Cn → Cn,

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is by definition the determinant of the N × N matrix of the

Related Topics:
Determinant - Matrix

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:Fij,

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where Fij is the partial derivative of Fi with respect to Xj.

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The condition

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:J ≠ 0

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enters into the inverse function theorem in multivariable calculus. In fact that condition for smooth functions (and so a fortiori for polynomials) ensures the existence of a local inverse function to F, at any point where it holds.

Related Topics:
Inverse function theorem - Multivariable calculus

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On the other hand in the polynomial case J is itself a polynomial. Since the complex numbers form an algebraically closed field J will be zero for some complex values of X1, ?, XN, unless we have the condition

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:J is a constant.

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Therefore it is a relatively elementary fact that

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:if F has an inverse function defined everywhere, then J is a constant.

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The Jacobian conjecture is the converse: it states that

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:if J is a non-zero constant function, then F has an inverse function.

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A proof for the two-variable case was announced in 2004 by Carolyn Dean, and has been submitted for journal publication. Several sources have reported that her proof contains an error. A series of talks which she scheduled have been cancelled. See, for example,

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http://mathworld.wolfram.com/JacobianConjecture.html.

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