  Let $R$ be a real closed field. We consider basic semi-algebraic sets defined
by $n$-variate equations/inequalities of $s$ symmetric polynomials and an
equivariant family of polynomials, all of them of degree bounded by $2d < n$.
Such a semi-algebraic set is invariant by the action of the symmetric group. We
show that such a set is either empty or it contains a point with at most $2d-1$
distinct coordinates. Combining this geometric result with efficient algorithms
for real root finding (based on the critical point method), one can decide the
emptiness of basic semi-algebraic sets defined by $s$ polynomials of degree $d$
in time $(sn)^{O(d)}$. This improves the state-of-the-art which is exponential
in $n$. When the variables $x_1, \ldots, x_n$ are quantified and the
coefficients of the input system depend on parameters $y_1, \ldots, y_t$, one
also demonstrates that the corresponding one-block quantifier elimination
problem can be solved in time $(sn)^{O(dt)}$.
