JacobianConjecture 2003-01-28 13:23:15 2006-09-15 17:03:14 Conjecture % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $F \colon \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial map, i.e., $$F(x_1,\dots,x_n) = (f_1(x_1,\dots,x_n), \dots,f_n(x_1,\dots,x_n))$$ for certain polynomials $f_i \in \mathbb{C}[X_1,\dots,X_n]$. If $F$ is invertible, then its Jacobi determinant $\det(\partial f_i/\partial x_j)$, which is a polynomial over $\mathbb{C}$, vanishes nowhere and hence must be a non-zero constant. The \emph{Jacobian conjecture} asserts the converse: every polynomial map $\mathbb{C}^n \to \mathbb{C}^n$ whose Jacobi determinant is a non-zero constant is invertible.