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Conditions are found on an operation on a finite set for its transform to be lacunary, that is, missing many expected terms. Often the condition is that the operation preserves a relation. General operations are split into odd and even lacunary parts, or more generally, into several lacunary parts given by a non-binary parity. This is applied to classical Fourier transforms as well as algebraic normal forms. With this theory, an explicit polynomial expansion is given for any operation in a preprimal algebra based on a finite elementary Abelian group. Convolution is defined, with a criterion given for it to commute with transforms. [ABSTRACT FROM AUTHOR]

The n th cyclotomic polynomial Φ n (X) is the minimal polynomial of ζ n : = e 2 π i / n. Given an integer m ≥ 1 and a prescribed set S of arithmetic progressions modulo m , we define n x as the product of the primes p ≤ x lying in those progressions. Let d (n) denote the number of divisors of n. It turns out that under certain conditions on S and m there exists j x such that log ⁡ | Φ n x (ζ m j x ) | / d (n x) tends to a positive limit. Our aim is to determine those conditions. We use the arithmetic of cyclotomic number fields, non-standard properties of character tables of finite abelian groups and a recent theorem of Bzdȩga, Herrera-Poyatos and Moree. After developing some generalities, we restrict to the case where m is a prime. Our motivation comes from a paper of Vaughan (1975). He studied the case where S = { ± 2 (mod 5) } and used it to show that the maximum coefficient in absolute value of Φ n can be very large. [ABSTRACT FROM AUTHOR]

Let t ∈ N + be given. In this article, we are interested in characterizing those d ∈ N + such that the congruence 1 t ∑ s = 0 t - 1 n + d ζ t s d - 1 ≡ n d - 1 (mod d) is true for each n ∈ Z . In particular, assuming that d has a prime divisor greater than t, we show that the above congruence holds for each n ∈ Z if and only if d = p r , where p is a prime number greater than t and r ∈ { 1 , ... , t } . [ABSTRACT FROM AUTHOR]

For the handlebody H g of genus g , Przytycki studied the (Kauffman bracket) skein module 𝒮 q (H n # H m) of the connected sum H n # H m at q. One of his results is that, in the case when 1 − q k is invertible for any k ≠ 0 , a homomorphism φ : 𝒮 q (H n ⊔ H m) → 𝒮 q (H n # H m) is an isomorphism, which is induced by a natural way. In this paper, in the case when n = m = 1 , the ground ring is ℂ , and q ∈ ℂ is a 4 k -th root of unity (k ≥ 2), we show that φ is not injective. [ABSTRACT FROM AUTHOR]

Let be an algebraic number of degree . We consider the set of positive integers such that the primitive th root of unity is expressible as a quotient of two conjugates of over . In particular, our results imply that is small. We prove that , where for each sufficiently large . We also show that, in terms of , this estimate is best possible up to a constant, since the constant cannot be replaced by any number smaller than . [ABSTRACT FROM AUTHOR]

We study non-commutative digit expansions in quaternion rings that arise in the context of supersingular elliptic curves. These digit expansions can be used in a $${\tau}$$ -and-add method to speed up arithmetic (scalar multiplication and pairing) on certain families of supersingular elliptic curves in characteristic $${p \geqq 5}$$ . The basis $${\tau}$$ is a quadratic algebraic integer that represents the Frobenius endomorphism of the curve, which is a very fast operation to evaluate. We prove the existence of a finite expansion for every element of the quaternion ring, as well as the equivalence between right and left digit expansions (i.e. the basis $${\tau}$$ is placed right, resp. left, to the digit); this expansion turns out to be a non-adjacent form (NAF) for integers, i.e. in every two consecutive digits there is at least one 0. [ABSTRACT FROM AUTHOR]

Any finite-dimensional Hopf algebra is Frobenius and the stable category of -modules is triangulated monoidal. To -comodule algebras we assign triangulated module-categories over the stable category of -modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable , our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds. [ABSTRACT FROM AUTHOR]

In order to study a class of finite-dimensional representations of U( sl), we deal with the quotient algebra U( m, n, b) of quantum group U( sl) with relations K = 1, E = b, F = 0 in this paper, where q is a root of unity. The algebra U( m, n, b) is decomposed into a direct sum of indecomposable (left) ideals. The structures of indecomposable projective representations and their blocks are determined. [ABSTRACT FROM AUTHOR]

Let K be a number field. For f∈K[x], we give an upper bound on the least positive integer T=T(f) such that no quotient of two distinct Tth powers of roots of f is a root of unity. For each ε>0 and each f∈ℚ[x] of degree d≥d(ε) we prove that $\log T(f)\lt (2+\varepsilon )\sqrt (d \log d)$. In the opposite direction, we show that the constant 2 cannot be replaced by a number smaller than 1 . These estimates are useful in the study of degenerate and nondegenerate linear recurrence sequences over a number field K. [ABSTRACT FROM AUTHOR]

Two methods to test whether a given polynomial has two distinct roots whose quotient is a root of unity are discussed. [ABSTRACT FROM AUTHOR]

We study the Yetter-Drinfeld D(B)-module algebra structure on the Heisenberg double H(B*) endowed with a 'heterotic' action of the Drinfeld double D(B). In terms of the braiding of Yetter-Drinfeld modules, H(B*) is braided commutative. By the Brzezinski-Militaru theorem, H(B*) # D(B) is then a Hopf algebroid over H(B*). The heterotic action can also be interpreted in the spirit of Lu's description of H(B*) as a twist of D(B). For B a particular Taft Hopf algebra at a 2pth root of unity, the construction is adapted to yield Yetter-Drinfeld module algebras over the 2p3-dimensional quantum group [image omitted]. In particular, it follows that Matp() is a braided commutative Yetter-Drinfeld [image omitted]-module algebra and [image omitted] is a Hopf algebroid over Matp(). [ABSTRACT FROM AUTHOR]

Abstract: The quantum Frobenius map and it splitting are shown to descend to maps between generalized q-Schur algebras at a root of unity. We also define analogs of q-Schur algebras for any affine algebra, and prove the corresponding results for them. [Copyright &y& Elsevier]

For an expanding integral s×s matrix A with &z.sfnc;detA&z.sfnc;=p, it is well known that if D={d0,…,dp−1}⊂Zs is a complete set of coset representatives of Zs/AZs, then T(A,D) is a self-affine tile. In this paper we show that if p is a prime, such D actually characterizes the tile digit sets provided that span(D)=Rs. This result is known for s=1, the one-dimensional case [R. Kenyon, in: Contemp. Math., vol. 135, 1992, pp. 239–264] and the question for s>1 has been considered by Lagarias and Wang [J. London Math. Soc. 53 (1996) 21–49] under some other conditions. The proof here involves a new setup to study the zeros of the mask m(ξ)=p−1∑j=0p−1e2πi〈ξ,dj〉. It can also be generalized to consider the existence of a compactly supported L1-solution of the refinement equation (scaling function) with positive coefficients. [Copyright &y& Elsevier]

We show that a theorem obtained by K. R. Slavin can be easily deduced from the q-Lucas theorem. [ABSTRACT FROM AUTHOR]