A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:42:54Z http://mathoverflow.net/feeds/question/11904 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11904/a-polynomial-map-from-n-to-n-mapping-the-positive-orthant-onto-n A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n? Louis Deaett 2010-01-15T19:34:36Z 2010-01-15T19:57:05Z <blockquote> <p><b>Question:</b> Is there a polynomial map from ℝ<sup>n</sup> to ℝ<sup>n</sup> under which the image of the positive orthant (the set of points with all coordinates positive) is all of ℝ<sup>n</sup>?</p> </blockquote> <p>Some observations:</p> <p>My intuition is that the answer must be 'no'... but I confess my intuition for this sort of geometric problem is not very well-developed.</p> <p>Of course it is relatively easy to show that the answer is 'no' when n=1. (In fact it seems like a nice homework problem for some calculus students.) But I can't seem to get any traction for n>1.</p> <p>This feels like the sort of thing that should have an easy proof, but then I remember feeling that way the first time I saw the Jacobian conjecture... now I'm wary of statements about polynomial maps of ℝ<sup>n</sup>!</p> http://mathoverflow.net/questions/11904/a-polynomial-map-from-n-to-n-mapping-the-positive-orthant-onto-n/11905#11905 Answer by Mariano Suárez-Alvarez for A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n? Mariano Suárez-Alvarez 2010-01-15T19:41:53Z 2010-01-15T19:57:05Z <p>The map $z\in\mathbb C\mapsto z^4\in\mathbb C$, when written out in coordinates, is a polynomial map which sends the <em>closed</em> first quadrant to the whole of $\mathbb R^2$---and by considering cartesian products you get the same for $\mathbb R^{2n}=\mathbb C^n$.</p> <p><strong>Later:</strong> as observed in a comment by <a href="http://mathoverflow.net/users/622/charles-siegel" rel="nofollow">Charles</a>, this can be turned into a solution for the <em>open</em> quadrant by composing with a translation, as in $z\in\mathbb C\mapsto (z-z_0)^4\in\mathbb C$ with $z_0$ in the open first quadrant.</p>