Estimates for the probability that a system of random equations is solvable in a given set of vectors over the field ${GF}(3)$

Abstract

Let P n be the probability that a second order system of nonlinear random equations over the field GF(3) has a solution in a given set of vectors, where n is the number of unknowns in the system. A necessary and sufficient condition is found for P n → 0 as n → ∞. Some rates of convergence to zero are found and some applications are described.