What tools are available from real algebraic geometry, analysis and topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$ from local properties and maybe function values?


Surjectivity of $f:M_{1}\rightarrow\mathbb{R}^{d}$ ($M_{1}$ real smooth non-compact manifold) is in particular satisfied if the function is invertible. This can be checked locally using global inversion theorems that go back to Hadamard. It is sufficient if the Jacobian is invertible everywhere. However, if the Jacobian has singularities, surjectivity can be verified if the Jacbobian has only ordinary singular points (and the function maps at least one point to the components of $\mathbb{R}^d$ arising from the image of the singularities, compare Theorem 2.6 in Section 3 of ``A primer of nolinear analysis'' by Ambrosetti and Prodi). What is known if this condition fails? What other tools are available to check surjectivity if $f$ is analytic?

Real algebraic geometry:

What is known if $M_{1}$ is a real algebraic varity and $f$ is a polynomial with coefficients that are algebraic numbers? There are many results available for algebraically closed fields but in this case the field is $\mathbb{R}.$

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    $\begingroup$ For your example from analysis, don't you need to assume that your map $f$ is proper ? It seems to me that the inclusion map from an open subset of $\mathbb{R}^d$ to $\mathbb{R}^d$ is a counterexample. $\endgroup$ – Thomas Richard Dec 9 '14 at 15:33
  • $\begingroup$ Thanks for spotting. I forgot to put that Assumption. But my question hints at using only bits of global properties. Properness is one such property. $\endgroup$ – warsaga Dec 10 '14 at 0:07

Topology: If the image of $f$ is both open and closed. If $f$ is proper then the image is closed, but this is quite restrictive since then each fiber of $f$ is compact.
Analysis: If $f$ is a submersion and has closed image.
Real algebraic geometry: If $f$ is a polynomial submersion and has closed image. I suspect that any polynomial mapping on a closed semialgebaic subset of $\mathbb R^m$ has closed image, but I could not find any reference for this.


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