On Mon, Jul 9, 2012 at 9:52 PM, Mike Stay <metaweta@gmail.com> wrote:
On Mon, Jul 9, 2012 at 6:51 PM, Mike Stay <metaweta@gmail.com> wrote:
On Mon, Jul 9, 2012 at 6:50 PM, Mike Stay <metaweta@gmail.com> wrote:
Is there a multivariate polynomial whose image over R^n is strictly positive?
Sorry, "strictly the positive reals".
Or better: "exactly the strictly positive reals."
No. Add a point at infinity to R^n and to R, extend the multvariate polynomial by mapping the point at infinity to the point at infinity, and the resulting function is still compact. Since the range is compact, so is the image. So the image of the original polynomial is compact once the point at infinity is added, and so it cannot e the strictly positive reals, since infinity maps to infinity, not 0. Andy
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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