Existence of element $(x_0,y)$ in a set of common zeros for all $(x_0,y)$ satisfying system of inequalities

Question

Let $f_1,f_2,\cdots,f_n,g_1,g_2\cdots,g_m\in \mathbb{R}[x,y]$, then define the affine variety and semi-affine variety as follows: $V(f_1,f_2,\cdots,f_n):=\{(x,y)\in\mathbb{C}^2: f_1(x,y)=f_2(x,y)=\cdots=f_n(x,y)=0\}$ and $V_{\ge 0}(g_1,g_2\cdots,g_m):=\{(x,y)\in\mathbb{R}^2: g_1(x,y)\ge 0, g_2(x,y)\ge 0,\cdots,g_m(x,y)\ge 0\}$.

Let $G$ be the Gröbner basis of the ideal $(f_1,f_2,\cdots,f_n)$ in $\mathbb{C}[x,y]$.

It is well-known that $V(f_1,f_2,\cdots,f_n)\neq\emptyset\iff G$ does not contain $1$.

What is the sufficient and necessary conditions in terms of Gröbner basis for the statement:

There exists $x_0\in\mathbb{R}$ such that $(x_0,y)\in V(f_1,f_2,\cdots,f_n)$ for all $(x_0,y)\in V_{\ge 0}(g_1,g_2\cdots,g_m)$.

Other computational ways to characterize the above statement would be welcome.

Answer 1

I did not check carefully, but it looks like if one constructs the Gröbner basis $H$ of $(g_1(x,y) - z_1^2, \dots,g_m(x,y) - z_m^2)$ with respect to any ordering "eliminating" $y$, then the necessary condition is that every $f_i(x,y)$ reduces w.r.t. $H$ to a polynomial that does not involve $y$.