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?

**Analysis**

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 $\mathbb{R}.$