  We design a new algorithm for solving parametric systems having finitely many
complex solutions for generic values of the parameters. More precisely, let $f
= (f_1, \ldots, f_m)\subset \mathbb{Q}[y][x]$ with $y = (y_1, \ldots, y_t)$ and
$x = (x_1, \ldots, x_n)$, $V\subset \mathbb{C}^{t+n}$ be the algebraic set
defined by $f$ and $\pi$ be the projection $(y, x) \to y$. Under the
assumptions that $f$ admits finitely many complex roots for generic values of
$y$ and that the ideal generated by $f$ is radical, we solve the following
problem. On input $f$, we compute semi-algebraic formulas defining
semi-algebraic subsets $S_1, \ldots, S_l$ of the $y$-space such that
$\cup_{i=1}^l S_i$ is dense in $\mathbb{R}^t$ and the number of real points in
$V\cap \pi^{-1}(\eta)$ is invariant when $\eta$ varies over each $S_i$.
  This algorithm exploits properties of some well chosen monomial bases in the
algebra $\mathbb{Q}(y)[x]/I$ where $I$ is the ideal generated by $f$ in
$\mathbb{Q}(y)[x]$ and the specialization property of the so-called Hermite
matrices. This allows us to obtain compact representations of the sets $S_i$ by
means of semi-algebraic formulas encoding the signature of a symmetric matrix.
When $f$ satisfies extra genericity assumptions, we derive complexity bounds on
the number of arithmetic operations in $\mathbb{Q}$ and the degree of the
output polynomials. Let $d$ be the maximal degree of the $f_i$'s and $D =
n(d-1)d^n$, we prove that, on a generic $f=(f_1,\ldots,f_n)$, one can compute
those semi-algebraic formulas with $O^~( \binom{t+D}{t}2^{3t}n^{2t+1}
d^{3nt+2(n+t)+1})$ operations in $\mathbb{Q}$ and that the polynomials involved
have degree bounded by $D$.
  We report on practical experiments which illustrate the efficiency of our
algorithm on generic systems and systems from applications. It allows us to
solve problems which are out of reach of the state-of-the-art.
