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Randomized black box algorithms provide a very e cient means for solving sparse linear systems over arbitrary elds. However, when these probabilistic algorithms fail, it is not revealed whether no solution exists or whether the algorithm simply made unlucky random choices. Here we give an e cient algorithm to compute a certi cate of inconsistency for a black box linear system over a eld. Our method requires a black box for the transpose of the matrix. The cost of producing the certi cate is shown to be about the same as that of solving the system in the black box model, while the cost of applying a given certi cate to prove inconsistency is much smaller. We also give an e cient algorithm for certifying that a sparse Diophantine linear system of integer equations has no integer solutions, even when it may have rational solutions.

Certifying inconsistency of sparse linear systems