http://openproblemgarden.org/op/jacobian_conjecture Comments for "Jacobian Conjecture" en http://openproblemgarden.org/op/jacobian_conjecture#comment-93661 <p>iff the determinant of the Jacobian matrix is a nonzero constant.</p> Thu, 04 Mar 2021 17:44:19 +0100 Anonymous comment 93661 at http://openproblemgarden.org http://openproblemgarden.org/op/jacobian_conjecture <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/keller_otto_heinrich">Keller</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/geometry">Geometry</a> » <a href="/category/algebraic_geometry">Algebraic Geometry</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png" alt="$ k $" /> be a field of characteristic zero. A collection <img class="teximage" src="/files/tex/c4cbeed48b0c6ce2a6294c75261da86958fb7393.png" alt="$ f_1,\ldots,f_n $" /> of polynomials in variables <img class="teximage" src="/files/tex/aaa29ecce30762f5fa8655c6d5de69088530a526.png" alt="$ x_1,\ldots,x_n $" /> defines an automorphism of <img class="teximage" src="/files/tex/b2baee4a1dacdb8cbb7880c33194a814fe14947b.png" alt="$ k^n $" /> if and only if the Jacobian matrix is a nonzero constant. </div> </tr></td> </table> </td> </tr> </table> Keller, Otto-Heinrich Affine Geometry Automorphisms Polynomials Geometry Algebraic Geometry http://openproblemgarden.org/op/jacobian_conjecture#comment Sun, 06 Jul 2008 05:58:52 +0200 Charles 1768 at http://openproblemgarden.org