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Revision difference : Jacobian conjecture |
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Version 1 |
| Let $F \colon \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial map, i.e., |
Let $F \colon \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial map, i.e., |
| $$F(x_1,\dots,x_n) = (f_1(x_1,\dots,x_n), \dots,f_n(x_1,\dots,x_n))$$ |
$$F(x_1,\dots,x_n) = (f_1(x_1,\dots,x_n), \dots,f_n(x_1,\dots,x_n))$$ |
| for certain polynomials $f_i \in \mathbb{C}[X_1,\dots,X_n]$. |
for certain polynomials $f_i \in \mathbb{C}[X_1,\dots,X_n]$. |
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| If $F$ is invertible, then its Jacobi determinant $\det(\partial f_i/\partial |
If $F$ is invertible, then its Jacobi determinant $\det(\partial f_i/\partial |
| x_j)$, which is a polynomial over $\mathbb{C}$, |
x_j)$, which is a polynomial over $\mathbb{C}$, |
| vanishes nowhere and hence must be a non-zero constant. |
vanishes nowhere and hence must be a non-zero constant. |
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The \emph{Jacobian conjecture} asserts the converse: every polynomial map
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The Jacobian conjecture asserts the converse: every polynomial map
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| $\mathbb{C}^n \to \mathbb{C}^n$ whose Jacobi determinant is a non-zero constant |
$\mathbb{C}^n \to \mathbb{C}^n$ whose Jacobi determinant is a non-zero constant |
| is invertible. |
is invertible. |
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