Cyclotomic Completions of Polynomial Rings

By Kazuo Habiro. 2002

68 citations recorded by the source corpus.

The main object of study in this paper is the completion Z[q]^N=\varprojlim_n Z[q]/((1-q)(1-q^2)...(1-q^n)) 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 sl_2 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-\zeta for each root \zeta of 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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/613a1aa6592d7d45874209068aef28bc0efa36ec

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Citations in the corpus

Mapping F_1-land:An overview of geometries over the field with one element (2009, background): “Shai Haran (cf. [20]) proposed using certain categories modeled over finite sets as a replacement for rings, and actually produced a candidate for the compactificationSpec Z of Spec Z in his framework.”
Bottom tangles and universal invariants (2005, background): “Hence we can naturally identify B(0, n) with the set BTn of isotopy classes of n– component bottom tangles.”
Chern-Simons theory, analytic continuation and arithmetic (2007, methodology): “unity. This way of thinking is motivated by the following features of the Habiro ring, shown in [Ha1, Ha2 ]: (a) It is easy to see that every element f(q) 2 \ Z[q±] can be written (nonuniquely) in the form:”
A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres (2006, background): “The invariant JM of an integral homology sphere M which we construct in the paper takes values in a completion Ẑ[q] of the polynomial ring Z[q], which was introduced in [17] and studied in [18].”