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

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.

• Sorry. I just forgot to type the $x$-component. Commented Nov 30, 2019 at 2:19

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 $$y$$.