Some lower bounds for the complexity of the linear programming feasibility problem over the reals

We analyze the arithmetic complexity of the linear programming feasibility problem over the reals. For the case of polyhedra defined by 2n2n half-spaces in Rn we prove that the set I(2n,n)I(2n,n), of parameters describing nonempty polyhedra, has an exponential number of limiting hypersurfaces  . From this geometric result we obtain, as a corollary, the existence of a constant c>1c>1 such that, if dense or sparse representation is used to code polynomials, the length of any quantifier-free formula expressing the set I(2n,n)I(2n,n) is bounded from below by Ω(cn)Ω(cn). Other related complexity results are stated; in particular, a lower bound for algebraic computation trees based on the notion of limiting hypersurface is presented.