The problem of answering Why-Not questions consists in explaining why the result of a query does not contain some expected data, i.e., missing answers. To solve this problem, we resort to identifying where in the query, data relevant to the missing answer were lost. Existing algorithms producing such query-based explanations rely on a query tree representation, potentially leading to different or partial explanations. This significantly impairs on the effectiveness of computed explanations. Here we present an effective, query-tree independent representation of query-based explanations, for a wide class of Why-Not questions, based on provenance polynomials. We further describe an algorithm that efficiently computes the complete set of these explanations. An experimental evaluation validates our statements