I am trying to solve a problem from Chapter 5 ("Degrees, Intersection Numbers and the Euler characteristic") of Hirsch's book "Differential topology". It goes like this:
" Let $f_1,\dots,f_n$ be real polynomials in $n\geq 2$ variables. Write $f_k=h_k + r_k$ where $h_k$ is a homogeneous polynomial of degree $d_k\geq 2$ and $r_k$ has smaller degree. Assume that $x=(0,\dots,0)$ is the only solution to $h_1(x)=\dots=h_n(x)=0$. Assume also that $\det(\frac{\partial h_i}{\partial x_j}(x))\neq 0$ at all nonzero $x\in \mathbb{R}^n$. Then the system of equations $f_1=\dots=f_n=0$ has a solution in $\mathbb{R}^n$.
Hint: Use the previous exercise."
I have done the previous exercise. It states that if $f: U\subseteq \mathbb{R}^n \to \mathbb{R^n}$ is a proper $\mathcal{C}^1$ map such that $\det(Df_x)$ does not change sign outside some compact set (and is not identically zero), then $f$ is surjective.
I am not looking for a solution to the exercise, but a good hint on how to use this lemma would be really appreciated.
I have tried to properly modify the morphism $f=(f_1,\dots,f_n)$ ( and $h$ ) into a proper one but didn't succeed.
First of all, thank you for the very useful comments.
The hint of user90189 on how to prove that $h$ is proper works. With a very similar argument you can prove that $f$ is also proper, so the degree of these functions is well defined.
If you apply the lemma to $h$ and use the proper homotopy $H(x,t)=h(x)+tr(x)$ to show that $f$ and $h$ have the same degree, you get that the degree of $f$ is not equal to zero, so it must be surjective.