We prove a version for motivic cohomology of Thomason’s theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H(k, Z/n(1)) corresponding to a primitive nth root of unity.

The following twelve poster abstracts were presented at the ANTS-8 poster session.1 ANTS-8 was held at the Banff Centre in Banff, Alberta Canada, May 17–22, 2008. The conference website, where many of the posters can be viewed online, is http://ants.math. ucalgary.ca/. Calculating Really Big Cyclotomic Polynomials Andrew Arnold and Michael Monagan, Simon Fraser University, ada26@ sfu.ca The nth...

The following twelve poster abstracts were presented at the ANTS-8 poster session.1 ANTS-8 was held at the Banff Centre in Banff, Alberta Canada, May 17–22, 2008. The conference website, where many of the posters can be viewed online, is http://ants.math. ucalgary.ca/. Calculating Really Big Cyclotomic Polynomials Andrew Arnold and Michael Monagan, Simon Fraser University, ada26@ sfu.ca The nth...

When G has size n and g ∈ G, for any character χ of G we have χ(g)n = χ(gn) = χ(1) = 1, so the values of χ lie among the nth roots of unity in S1. More precisely, the order of χ(g) divides the order of g (which divides #G). Characters on finite abelian groups were first studied in number theory, since number theory is a source of many interesting finite abelian groups. For instance, Dirichlet u...

We give upper and lower bounds on the count of positive integers n x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x) = (x n − 1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n < 561 all coefficients of Ψn(x) are ≤ 1 in absolute value. We establish variou...

In this article we develop a general method by which one can explicitly evaluate certain sums of nth powers products d≥1 elementary trigonometric functions evaluated at m=(m1,…,md)-th roots unity. Our approach is to first identify the individual terms in expression under consideration as eigenvalues discrete Laplace operator associated graph whose vertices form d-dimensional torus Gm depends on...

Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + xZ x ) can be written as f = gn for g ∈ R, n 2. Let Pn := {gn | g ∈R} and let μn := n∏p |n p. We show among other things that (i) for f ∈R, f ...