Cyclotomic completions of polynomial rings

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The main object of study in this paper is the completion Z[q]ⁿ=ₙ Z[q]/((1-q)(1-q²)...(1-qⁿ)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3-spheres with values in Z[q]ⁿ announced by the author, which unifies all the sl₂ Witten-Reshetikhin-Turaev invariants at various roots of unity. We show that any element of Z[q]ⁿ is uniquely determined by its power series expansion in q-ζfor each root ζof unity. We also show that any element of Z[q]ⁿ is uniquely determined by its values at the roots of unity. These results may be interpreted that Z[q]ⁿ behaves like a ring of ``holomorphic functions defined on the set of the roots of unity''. We will also study the generalizations of Z[q]ⁿ, which are completions of the polynomial ring R[q] over a commutative ring R with unit with respect to the linear topologies defined by the principal ideals generated by products of powers of cyclotomic polynomials.