Closed paths whose steps are roots of unity
نویسندگان
چکیده
We give explicit formulas for the number Un(N) of closed polygonal paths of length N (starting from the origin) whose steps are n roots of unity, as well as asymptotic expressions for these numbers when N → ∞. We also prove that the sequences (Un(N))N≥0 are P -recursive for each fixed n ≥ 1 and leave open the problem of determining the values of N for which the dual sequences (Un(N))n≥1 are P -recursive. Résumé. Nous donnons des formules explicites pour le nombre Un(N) de chemins polygonaux fermés de longueur N (débutant à l’origine) dont les pas sont des racines n-ièmes de l’unité, ainsi que des expressions asymptotiques pour ces nombres lorsque N → ∞. Nous démontrons aussi que les suites (Un(N))N≥0 sont P -récursives pour chaque n ≥ 1 fixé et laissons ouvert le problème de déterminer les valeurs de N pour lesquelles les suites duales (Un(N))n≥1 sont P -récursives.
منابع مشابه
Galois Extensions of Lubin-tate Spectra
Let En be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E n whose coefficients are built from the coefficients of En and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that the...
متن کاملModels of some genus one curves with applications to descent
Let p denote a prime, and K a field of characteristic prime to p and containing the pth roots of unity. For p equal to 3 and 5, the author finds a scheme Tp and a family of genus one curves over Tp such that any genus one curve defined over the field K of index p whose Jacobian elliptic curve E has E[p](K) = E[p](K) is isomorphic to a curve lying over a Kpoint of Tp . The author then relates th...
متن کاملExamples of non-trivial roots of unity at ideal points of hyperbolic 3-manifolds
This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character varieties have ideal points for which the associated roots of unity are not ±1. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to whether roots of unity other than ±1 occur.
متن کاملRoots of unity as a Lie algebra
This note gives a categorical development arising from a theorem of A.A. Klyachko relating the Lie operad to roots of unity. We examine the "substitude" structure on the groupoid C whose homsets are the cyclic groups. The roots of unity representations of the cyclic groups form a Lie algebra for a certain oplax monoidal structure on the category of linear representations of C . Mathematics Subj...
متن کاملOn the Irreducible Representations of a Class of Pointed Hopf Algebras
We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work of N. Andruskiewitsch and the second author. Special cases are the multiparameter deformations of the enveloping algebras of semisimple Lie algebras where th...
متن کامل