A well-characterized approximation problem

We consider the following NP optimization problem: Given a set of polynomials P_i(x), i = 1,...,s, of degree at most 2 over GF[p] in n variables, find a root common to as many as possible of the polynomials P_i(x). We prove that in the case in which the polynomials do not contain any squares as monomials, it is always possible to approximate this problem within a factor of p²/(p-1) in polynomial time for fixed p. This follows from the stronger statement that one can, in polynomial time, find an assignment that satisfies at least (p-1)/p² of the nontrivial equations. More interestingly, we prove that approximating the maximal number of polynomials with a common root to within a factor of p - ε is NP-hard. This implies that the ratio between the performance of the approximation algorithm and the impossibility result is essentially p /(p-1), which can be made arbitrarily close to 1 by choosing p large. We also prove that for any constant δ < 1, it is NP-hard to approximate the solution of quadratic equations over the rational numbers, or over the reals, within n^δ.