Abstract
These inequalities are concerned with the distribution of zeros of polynomial congruences in incomplete residue systems modulo an integer m≥2. If B denotes a subset of a complete system C , the difference between the number N(f; B ) of zeros x ∈ B of f ∈ Z [ x] and the expected quota | B | C | −1 N(f; C ), where | · | denotes cardinality, is bounded. In certain applications it is useful to have similar bounds when multiplicities are taken into account i.e., when more than one representative from a residue class is allowed. The modifications for this are provided, together with an interpretation in terms of the Macbeath region. For quadratic polynomials, the relative strengths of these inequalities (as well as the original ones) are shown to be comparable.
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