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We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over _2 each of which is a product of affine forms. We focus on the case of k-CNF formulas (the k-SUB-SAT problem). Clearly, it is no easier than k-SAT, and might bearXiv:2108.05914v1 fatcat:btn7airnijg47h5xkaiplandoa