Minimal achievable approximation ratio for MAX-MQ in finite fields

Given a multivariate quadratic polynomial system in a finite field Fq, the problem MAX-MQ is to find a solution satisfying the maximal number of equations. We prove that the probability of a random assignment satisfying a non-degenerate quadratic equation is at least 1q −O(q− n 2 ), where n is the number of the variables in the equation. Consequently, the random assignment provides a polynomial-time approximation algorithm with approximation ratio q + O(q− n 2 ) for non-degenerate MAX-MQ. For large n, the ratio is close to q. According to a result by H̊astad, it is NP-hard to approximate MAX-MQ with an approximation ratio of q− 2 for a small positive number 2. Therefore, the minimal approximation ratio that can be achieved in polynomial time for MAX-MQ is q.