Let $\Omega$ be a non-empty set in ${\mathbb R}^n$ defined by a set of polynomial inequalities with rational coefficients $P_i(x_1, \ldots ,x_n) \gt 0 (1 \leq i\leq m)$ and $Q_j(x_1, \ldots ,x_n) \geq 0 (1 \leq j\leq m')$. Let $F(x_1, \ldots ,x_n)$ be a polynomial with rational coefficients such that $F$ attains its minimum on $\Omega$ at a unique point $(z_1, \ldots ,z_n)$. Does it follow that all the $z_k$ are algebraic over $\mathbb Q$ ?
UPDATE (15 mins later) :since an answer by André Nicolas pointed out that this is a standard fact in model theory, I guess I should rephrase my question as, "Where may I learn more about this" (preferably on the internet ?)