  A beautiful result of Br\"ocker and Scheiderer on the stability index of
basic closed semi-algebraic sets implies, as a very special case, that every
$d$-dimensional polyhedron admits a representation as the set of solutions of
at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case,
however, no constructive proof is known, even if the quadratic upper bound is
replaced by any bound depending only on the dimension.
  Here we give, for simple polytopes, an explicit construction of polynomials
describing such a polytope. The number of used polynomials is exponential in
the dimension, but in the 2- and 3-dimensional case we get the expected number
$d(d+1)/2$.
