Unsatisfiable systems of equations, over a finite field.

The properties of any system of k simultaneous equations in n variables over GF(q), are studied, with a particular emphasis on unsatisfiable systems. A general formula for the number of solutions is given, which can actually be useful for computing that number in the special case here all the equations are of degree 2. When such a quadratic system hers no solution, there is always a proof of unsatisfiability of size q(n/2) times a polynomial in n and q, which can be checked deterministically in time satisfying a similar bound. Such a proof can be found by a probabilistic algorithm in time asymptotic to that required to test, by substitution in k quadratic equations, all q(n) potential solutions.