Minimal polynomial descriptions of polyhedra and special semialgebraic sets

We show that a $d$-dimensional polyhedron $S$ in ${\mathrm{\Re }}^d$ can be represented by $d$-polynomial inequalities, that is, $S = {x \in \real^d : p_0(x) \ge 0,

…, p_{d-1}(x) \ge 0 }$,where$p_0,…,p_{d-1}$ are appropriate polynomials. Furthermore, if an elementary closed semialgebraic set $S$ is given by polynomials $q_1,\mathrm{.}\mathrm{.}\mathrm{.},q_k$ and for each $x\in S$ at most $s$ of these polynomials vanish in $x$, then $S$ can be represented by $s+1$ polynomials (and by $s$ polynomials under the extra assumption that the number of points $x \in S$inwhich$s$ $q_i$’s vanish is finite).