  With every real polynomial $f$, we associate a family $\{f_{\epsilon
r}\}_{\epsilon, r}$ of real polynomials, in explicit form in terms of $f$ and
the parameters $\epsilon>0,r\in N$, and such that $\Vert f-f_{\epsilon
r}\Vert_1\to 0$ as $\epsilon\to 0$.
  Let $V\subset R^n$ be a real algebraic set described by finitely many
polynomials equations $g_j(x)=0,j\in J$, and let $f$ be a real polynomial,
nonnegative on $V$. We show that for every $\epsilon>0$, there exist
nonnegative scalars $\{\lambda_j(\epsilon)\}_{j\in J}$ such that, for all $r$
sufficiently large,
  $$f_{\epsilon r}+\sum_{j\in J} \lambda_j(\epsilon) g_j^2,\quad is a sum of
squares.$$
  This representation is an obvious certificate of nonnegativity of
$f_{\epsilon r}$ on $V$, and very specific in terms of the $g_j$ that define
the set $V$. In particular, it is valid with {\it no} assumption on $V$. In
addition, this representation is also useful from a computation point of view,
as we can define semidefinite programing relaxations to approximate the global
minimum of $f$ on a real algebraic set $V$, or a semi-algebraic set $K$, and
again, with {\it no} assumption on $V$ or $K$.
