Satisfying Degree-d Equations over GFn
We study the problem where we are given a system of polynomial equations defined by multivariate polynomials over GF of fixed constant degree d > 1 and the aim is to satisfy the maximal number of equations. A random assignment approximates this problem within a factor 2− d and we prove that for any ε > 0, it is NP-hard to obtain a ratio 2− d + ε. When considering instances that are perfectly satisfiable we give a probabilistic polynomial time algorithm that, with high probability, satisfies a fraction 21 − d − 21 − 2d and we prove that it is NP-hard to do better by an arbitrarily small constant. The hardness results are proved in the form of inapproximability results of Max-CSPs where the predicate in question has the desired form and we give some immediate results on approximation resistance of some predicates.