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Home > Jacobian conjecture

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

X₁, … , X_N,

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

F: Cⁿ → Cⁿ,

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

F_ij,

where F_ij is the partial derivative of F_i with respect to X_j.

The condition

J ≠ 0

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.

On the other hand in the polynomial case J is itself a polynomial. Since the complex numbers form an algebraically closed fieldAbstract algebra In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F has a zero in F''. In that case, every such polynomial splits into linear factors. It can be shown that a field J will be zero for some complex values of X₁, … , X_N, unless we have the condition

J is a constantIn mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed. Constant number The most widely mentioned sort of constant is a fixed, but possibly unspecified, n.

Therefore it is a relatively elementary fact that

if F has an inverse functionIn mathematical analysis, an inverse function is in simple terms a function which "does the reverse" of a given function. More formally, if f is a function with domain X, then f -1 is its inverse function if and only if for every we have: :. For example, defined everywhere, then J is a constant.

The Jacobian conjecture is the converseConverse Shoes is an American shoe company. In logic, each implicational statement has a corresponding converse.: it states that

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

A proof for the two variable case was announced in 2004 by Carolyn Dean , and has been submitted for journal publication.

External link

• Web page of T. T. Moh on the conjecture

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Topics: Jacobian Conjecture Function Polynomials Inverse Complex...

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JacobianIn vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant . Also, in algebraic geometry the Jacobian of a curve means the Jacobian variety: a group structure, which can be imposed on the curv Jacobian conjectureIn 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