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To compute solutions of sparse polynomial systems efficiently we have toexploit the structure of their Newton polytopes. While the application ofpolyhedral methods naturally excludes solutions with zero components, anirreducible decomposition of a variety is typically understood in affine space,including also those components with zero coordinates. We present a polyhedralmethod to compute all affine solution sets of a polynomial system. The methodenumerates all factors contributing to a generalized permanent. Toric solutionsets are recovered as a special case of this enumeration. For sparse systems asadjacent 2-by-2 minors our methods scale much better than the techniques fromnumerical algebraic geometry.

A Polyhedral Method to Compute All Affine Solution Sets of Sparse Polynomial Systems