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We extend the Ax-Katz theorem for a single polynomial from finite fields tothe rings Z_m with m composite. This extension not only yields the analogousresult, but gives significantly higher divisibility bounds. We conjecture whatcomputer runs suggest is the optimal result for any m, and prove a special caseof it. The special case is for m = 2^r and polynomials of degree 2. Our resultsalso yield further properties of the solution spaces. Polynomials modulocomposites are the focus of some computational complexity lower boundfrontiers, while those modulo 2^r arise in the simulation of quantum circuits.We give some prospective applications of this research.

Polynomials Modulo Composite Numbers: Ax-Katz type theorems for the structure of their solution sets