Pseudo algebraically closed field

The field $K$ is pseudo algebraically closed if one of the following equivalent conditions holds:

Each absolutely irreducible variety $V$ defined over $K$ has a $K$ -rational point.
Each absolutely irreducible polynomial $f\in K[T_1,T_2,\cdots ,T_r,X]$ with $\frac{\partial f}{\partial X}\not =0$ and for each $0\not =g\in K[T_1,T_2,\cdots ,T_r,X]$ there exists $(\textbf{a},b)\in K^{r+1}$ such that $f(\textbf{a},b)=0$ and $g(\textbf{a})\not =0$ .
Each absolutely irreducible polynomial $f\in K[T,X]$ has infinitely many $K$ -rational points.
If $R$ is a finitely generated integral domain over $K$ with quotient field which is regular over $K$ , then there exist a homomorphism $h:R\to K$ such that $h (a) = a$ for each $a\in K$