Minimal polynomial descriptions of polyhedra and special semialgebraic sets

Abstract

We show that a $d$-dimensional polyhedron $S$ in ${\Re }^d$ can be represented by $d$-polynomial inequalities, that is, $S=\{x\in {\Re }^d:p_0(x)\ge 0,>...,p_{d-1}(x)\ge 0\}$, where $p_0,...,p_{d-1}$ are appropriate polynomials. Furthermore, if an elementary closed semialgebraic set $S$ is given by polynomials $q_1,...,q_k$ 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 $x\in S$ in which $s$ $q_i$'s vanish is finite).