Computing rational points in convex semi-algebraic sets and SOS decompositions

Let ${\cal P}=\{h_1, ..., h_s\}\subset \Z[Y_1, ..., Y_k]$, $D\geq \deg(h_i)$ for $1\leq i \leq s$, $\sigma$ bounding the bit length of the coefficients of the $h_i$'s, and $\Phi$ be a quantifier-free ${\cal P}$-formula defining a convex semi-algebraic set. We design an algorithm returning a rational point in ${\cal S}$ if and only if {\cal S}\cap \Q\neq\emptyset. It requires \sigma^{\bigO(1)}D^{\bigO(k^3)} bit operations. If a rational point is outputted its coordinates have bit length dominated by \sigma D^{\bigO(k^3)}. Using this result, we obtain a procedure deciding if a polynomial $f\in \Z[X_1, >..., X_n]$ is a sum of squares of polynomials in \Q[X_1, ..., X_n]. Denote by $d$ the degree of $f$, $\tau$ the maximum bit length of the coefficients in $f$, $D={{n+d}\choose{n}}$ and $k\leq D(D+1)-{{n+2d}\choose{n}}$. This procedure requires \tau^{\bigO(1)}D^{\bigO(k^3)} bit operations and the coefficients of the outputted polynomials have bit length dominated by \tau D^{\bigO(k^3)}