Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, this condition eventually forces the fibers to be "closed contours" about the origin.

Can this be generalized to a condition on the coefficients of a power series to ensure properness of an analytic mapping $\mathbb R^n\overset{f}{\to} \mathbb R$?

  • $\begingroup$ What exactly do you want to generalize: a) that the map is proper? or b) that fibers are bounded? These are two different properties since there can be infinitely many of fibers. $\endgroup$ – Alexandre Eremenko Feb 23 at 12:32
  • $\begingroup$ I'm interested in both, but properness is the first priority. $\endgroup$ – Arrow 2 days ago
  • $\begingroup$ If in the expansion in terms of homogeneous polynomials, all sufficiently large terms are positive then your map is proper. $\endgroup$ – Alexandre Eremenko 2 days ago

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