Showing papers on "Ring of integers published in 1996"
TL;DR: Two new lower bounds on the$MOD_{m}$-degree of the $MOD_{l}$ and $
eg MOD_{m]$ functions are proved, where $m$ is any composite integer and $l$ has a prime factor not dividing $m$.
Abstract: Define the $MOD_{m}$-degree of a boolean function $F$ to be the smallest degree of any polynomial $P$, over the ring of integers modulo $m$, such that for all 0-1 assignments $\vec{x}$, $F(\vec{x}) =0$ iff $P(\vec{x}) =0$. By exploring the periodic property of the binomial coefficients modulo $m$, two new lower bounds on the $MOD_{m}$-degree of the $MOD_{l}$ and $
eg MOD_{m}$ functions are proved, where $m$ is any composite integer and $l$ has a prime factor not dividing $m$. Both bounds improve from sublinear to $\Omega(n)$. With the periodic property, a simple proof of a lower bound on the $MOD_{m}$-degree with symmetric multilinear polynomial of the OR function is given. It is also proved that the majority function has a lower bound $n \over 2$ and the {MidBit} function has a lower bound $\sqrt{n}$.
39 citations
TL;DR: It is shown that 2k pk is a linear combination (over Z ) of multiplicative arithmetic functions and the sequence Cay( Z n, U n ) , where Z n is the ring of integers modulo n and Zn is the multiplicative group of unitsmodulo n has the map.
19 citations
TL;DR: In this paper, the second invariant of an extension of number fields defined by Chinburg via the canonical class of the extension and lying in the locally free class group was investigated, and it was shown that in Queyrut's S -class group, where S is a (finite) set of primes, the image of Chinburg's invariant equals the stable isomorphism of the ring of integers.
3 citations
TL;DR: In this paper, the number of m by n matrices with real rank r over the ring of integers modulo pk where p is a prime and k > 1 is computed.
3 citations
01 Jan 1996
TL;DR: In this paper, it was shown that under certain conditions (including Galois) a normset has unique factorization if and only if its corresponding ring of integers has a unique factorisation.
Abstract: In a recent paper by the author the normset and its multiplicative structure was studied. In that paper it was shown that under certain conditions (including Galois) that a normset has unique factorization if and only if its corresponding ring of integers has unique factorization. In this paper we shall examine some of the properties of a normset and describe what it says about the class group of the corresponding ring of integers.
2 citations
01 Jan 1996
TL;DR: Theorem 8 of [Q2] and Theoreme 1 of [S2] as mentioned in this paper shows that the localization exact sequence in algebraic K-theory splits into short exact sequences.
Abstract: Let F be a number field, O the ring of algebraic integers in F, and let θ denote the inclusion O ↪ F. The localization exact sequence in algebraic K-theory splits into short exact sequences
$$ 0 \to K_n O\xrightarrow{{\theta _\# }}K_n F \to \oplus _m K_{n - 1} \left( {O/m} \right) \to 0 $$
for all positive integers n, where θ # is the homomorphism induced by θ in K-theory and where m runs over the set of maximal ideals of O (see Section 5 of [Q1], Theorem 8 of [Q2] and Theoreme 1 of [S2]); in particular, θ # is always injective. On the other hand, G. Banaszak investigated the subgroup of divisible elements in K n F and explained the important role of these elements in relation with the Lichtenbaum-Quillen conjecture and etale K-theory (see [B1], [B2], [BG], [BZ]). If n is odd, K n F is a finitely generated abelian group and has therefore no non-trivial divisible elements. If n is even, K n F is a large torsion group but all its divisible elements belong to the image of θ # because ⊕ m K n-1 (O/m) is a direct sum of finite cyclic groups and hence contains no non-trivial divisible elements.
2 citations
TL;DR: In this article, Davenport et al. gave an affirmative answer under rather general conditions, and also new types of counter-examples, under the assumption that the polynomials in the ring of integers are the rationals.
2 citations
01 Jan 1996
TL;DR: In this paper, the local integrals Zm (t, x) = J x(det(x)) I det(x) I'dx M n(0C) where OC represents the integers of a composition algebra over a nonarchimedean local field K and X is a non-trivial character on the units in the ring of integers of K extended to K* by setting x(7r) = 1.
Abstract: This paper investigates the local integrals Zm (t, x) = J x(det(x)) I det(x) I'dx M n(0C) where OC represents the integers of a composition algebra over a non-archimedean local field K and X is a non-trivial character on the units in the ring of integers of K extended to K* by setting x(7r) = 1. The local zeta function for the trivial character is known for all composition algebras C. In this paper, we show in the quaternion case that Z(t, x) = 0 for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for X equal to the quadratic character. In this latter case, Z(t, X) = 0 for any character of order greater than 2.
18 Aug 1996
TL;DR: The aims of the paper are to provide and efficient algorithm for approximation of the real input signal with arbitrarily small error as an element of a quadratic number ring, and to prove the restrictions of the RNS moduli used in order to simplify the multiplication in the ring.
Abstract: Recent work has focused on doing residue computations that are quantization within a dense ring of integers in the real domain. The aims of the paper are to provide and efficient algorithm for approximation of the real input signal with arbitrarily small error as an element of a quadratic number ring, and to prove the restrictions of the RNS moduli used in order to simplify the multiplication in the ring. The proposed approximation scheme can be used for implementation of real-valued transforms and their multidimensional generalizations.