Properties of random systems of discrete equations with nonuniform sampling of unknowns

A random system $S$ of $M=M(n)$ discrete equations with n unkowns is considered. Each equation contains at most m unknowns which are selected at random, independently and (possibly) nonuniformly. We find conditions on the structure of equations ensuring that for $M-c\sqrt n=o(\sqrt n)$, $n\to\infty$, $m=\operatorname{const}$, the limit probability of solvability decreases continuously from $1$ to $0$ when $c$ increases from $0$ to $\infty$. An algorithm detecting the unsolvability of a random system of equations is described. This algorithm has low time complexity $O(\sqrt n)$. The limiting probability of detection the unsolvability is the same as for the exhaustive search algorithm. Our proofs are based on the geometric properties of random system of equations.