  Let $\xx_n=(x_1,\ldots,x_n)$ and $f\in \R[\xx_n,k]$. The problem of finding
all $k_0$ such that $f(\xx_n,k_0)\ge 0$ on $\mathbb{R}^n$ is considered in this
paper, which obviously takes as a special case the problem of computing the
global infimum or proving the semi-definiteness of a polynomial.
  For solving the problems, we propose a simplified Brown's CAD projection
operator, \Nproj, of which the projection scale is always no larger than that
of Brown's. For many problems, the scale is much smaller than that of Brown's.
As a result, the lifting phase is also simplified. Some new algorithms based on
\Nproj\ for solving those problems are designed and proved to be correct.
Comparison to some existing tools on some examples is reported to illustrate
the effectiveness of our new algorithms.
