Showing papers on "Ring of integers published in 1990"
TL;DR: In this article, an explicit estimate for the difference between the Weil height and the canonical height of points on elliptic curves is given in terms of the j-invariant and discriminant of the elliptic curve.
Abstract: Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit estimate for this difference in terms of the j-invariant and discriminant of the elliptic curve. The method of proof, suggested by Serge Lang, is to use the decomposition of the canonical height into a sum of local heights. We illustrate one use for our estimate by computing generators for the Mordell-Weil group in three examples. Let E be an elliptic curve defined over a number field K, say given by a Weierstrass equation (1) y2 =x +Ax+B with A and B in the ring of integers of K. The canonical height on E is a quadratic form h: E(K)-+ R. (For the definition and basic properties of h, see [10, Chapter VIII, ?9 or 6, Chapter VI].) The canonical height is determined by this property together with the fact that the difference
137 citations
Journal Article•
47 citations
TL;DR: In this paper, the problem of finding a minimal free resolution of S/I t, over arbitrary base ring R = Z, the ring of integers, has been studied in the Noetherian commutative ring setting.
Abstract: Let R be a Noetherian commutative ring with, unit element, and X ij be variables with 1 ≤ i ≤ m and 1 ≤ j ≤ n . Let S = R[x ij ] be the polynomial ring over R , and I t be the ideal in S , generated by the t × t minors of the generic matrix (x ij ) ∈ M m, n (S) . For many years there has been considerable interest in finding a minimal free resolution of S/I t , over arbitrary base ring R . If we have a minimal free resolution P. over R = Z, the ring of integers, then R′ ⊗ z P . is a resolution of S/I t over the base ring R′ .
31 citations
TL;DR: In this article, the system ℒ(H) = {L(a)|a eH/H× is studied, where L(a) =ka has a factorization into irreducibles of lengthk.
Abstract: LetH be a semigroup with divisor theory and finite divisor class groupG such that every class contains a prime divisor (e.g.H the multiplicative semigroup of the ring of integers of an algebraic number field). For an elementa eH / H× letL(a) =ka has a factorization into irreducibles of lengthk}. In this paper the system ℒ(H) = {L(a)|a eH / H× is studied.
22 citations
01 Mar 1990
TL;DR: In this paper, Heinzer showed that if D = Z is the ring of rational integers, then the ideal class group of R is a free abelian group on a countably infinite basis.
Abstract: Let D be a Dedekind domain and R = Int(D) be the ring of integer-valued polynomials of D. We relate the ideal class groups of D and R . In particular we prove that, if D = Z is the ring of rational integers, then the ideal class group of R is a free abelian group on a countably infinite basis. If D is an integral domain with field of fractions K, the ring of integervalued polynomials of D is denoted by Int(D) and is defined to be the subring of K[t] (where t is an indeterminate) consisting of those polynomials f(t) in K[t] such that f(D) C D. Work on rings of integer-valued polynomials has a long history. In particular, Int(7Z) , THE ring of integer-valued polynomials, has been studied at least since the work of Ostrowski [0] and Polya [P]. It was well known even then that Int(Z) is a free module over Z, with a basis consisting of the binomial polynomials Bo (t) , B1 (t), ... , where Bn (t) = t(t 1 ) . .. (t n + 1 )/n! Polya [P] gives a similar result with Z replaced by the ring of algebraic integers in a finite algebraic number field of class number 1. Polya showed that if the integral closure D of Z in a finite algebraic number field is of class number 1 , then Int(D) is a free D-module with a basis consisting of one polynomial of each nonnegative degree. He called such a D-basis for Int(D) a "regular basis". His proof applies for any principal ideal domain D of characteristic zero. Cahen [Ca, ?2] proved that if D is a Dedekind domain, Int(D) is a free D-module, and that if D is a principal ideal domain, then Int(D) has a regular basis. Received by the editors June 26, 1989. This research was presented to the 845th meeting of the AMS, October 28-29, 1988, Lawrence, Kansas, by Professor Heinzer. 1980 Mathematics Subject Classification (1985 Revision). Primary 13B25, 13F05; Secondary 12B05.
14 citations
TL;DR: It is shown how the use of a certain integral basis for cyclotomic fields enables one to perform the basic operations in their ring of integers efficiently.
Abstract: It is shown how the use of a certain integral basis for cyclotomic fields enables one to perform the basic operations in their ring of integers efficiently. In particular, from the representation with respect to this basis, one obtains immediately the smallest possible cyclotomic field in which a given sum of roots of unity lies. This is of particular interest when computing with the ordinary representations of a finite group.
12 citations
TL;DR: In this paper, a conjecture is formulated that gives necessary and sufficient conditions for the non-vanishing of the p -class group of the ring of integers in the maximal real subfield K + ( p ) of K ( p ).
12 citations
TL;DR: In this paper, an isomorphism of left R-modules between T and its associated graded ring R[xl,...,,, a polynomial ring over R is studied.
Abstract: Let R be a commutative algebra over the commutative ring k, and let A = {51...5, } be a finite set of commuting k-linear derivations from R to R. Let T=R[61I..6 'al'...,5-,n] be the corresponding ring of differential operators. We define and study an isomorphism of left R-modules between T and its associated graded ring R[xl,..., , a polynomial ring over R. This isomorphism is used to study the prime ideals of T, with emphasis on the question of catenarity. We prove that T is catenary when R is a commutative noetherian universally catenary k-algebra and one of the following cases occurs: (A) k is a field of characteristic zero and A acts locally finitely; (B) k is a field of positive characteristic; (C) k is the ring of integers, R is affine over k, and A acts locally finitely.
9 citations
TL;DR: In this article, the 2-primary subgroup of K2(OF), the Milnor K-group of the ring of integers of F, is of a particularly simple type, namely number fields that have the following property: F is totally real and K 2 (O F ) is elementary abelian of smallest rank possible.
6 citations
03 Jan 1990
TL;DR: It is shown that Newton's method provides polylog-time solutions for a broader class of linear systems than was previously supposed and can be implemented on a special-purpose architecture based on the three-dimensional mesh of trees.
Abstract: This thesis considers the mesh of trees architecture as both a special-purpose and a general-purpose parallel computer. A family of special-purpose VLSI architectures for computing an ($n\sb1 \times n\sb2 \times \cdots \times n\sb{d}$)-point multidimensional DFT over $\doubz\sb{M}$, the ring of integers modulo $M$, is proposed. Using the two-dimensional mesh of trees as a component, these architectures achieve optimal VLSI area $A$ = $\Theta((N\sp2\log\sp{2}M)/T\sp2)$ for any given computation time $T\ \epsilon$ ($\Omega(\log N),O(\sqrt{N\log M})\rbrack.$
The convergence properties of Newton's method are studied. By introducing and formalizing the notion of attunement of a linear system of equations, it is shown that Newton's method provides polylog-time solutions for a broader class of linear systems than was previously supposed. In particular, the system matrix need not be well-conditioned; all that is required is that the known vector be well-attuned to the system matrix. It is then shown that Newton's method can be implemented on a special-purpose architecture based on the three-dimensional mesh of trees. This same architecture can be used to construct the stiffness equations arising from a finite element approximation. Furthermore, it can be hybridized with a systolic array to achieve a processor-time or area-time tradeoff.
Then, in a different vein, the two-dimensional mesh of trees is studied as a general-purpose parallel computer. It is shown that this architecture can afford finer memory granularity and, thereby, reduce the memory redundancy required for deterministic P-RAM simulation. A distributed-memory, bounded-degree network model of parallel computation is proposed that allows one to take greater advantage of the potential for fine-grain memories without sacrificing other aspects of realism. The simulation scheme presented is admitted by this new model and achieves constant memory redundancy.
3 citations
03 Apr 1990
TL;DR: In this article, it was shown that a class of cubic integers provides a more efficient and less costly solution than other rings of integers which have been considered previously, and that certain quartic integer rings provide a simple solution to the problem of approximating roots of unity by using fields of lower degree.
Abstract: Recent work has focused on doing residue computations that use quantization within a particular dense ring of integers in the complex plane. That work is generalized, and it is shown that a class of cubic integers provides a more efficient and less costly solution than other rings of integers which have been considered previously. In addition, it is shown that certain quartic integer rings provide a simple solution to the problem of approximating roots of unity by using fields of lower degree. In addition, algorithms for approximating real and complex numbers with specific integer rings are developed. >
TL;DR: In this paper, the main object of this paper is to describe the structure of nil rings with minimum condition on principal right ideals (Theorem 1), where Z is the ring of integers.
Abstract: In what follows under a ring we always mean an associative one. The main object of this paper is to describe the structure of nil rings with minimum condition on principal right ideals (Theorem 1). Let Z be the ring of integers. The minimum condition for principal right ideals means that every descending chain of principal right ideals (al),D(a2),~(a8),D ... D terminates after a finite number of steps where (a~),=aiZ+atA. By MHR-ring (MHL-ring) we mean a ring with minimum condition on principal right (left) ideals. MHR-rings were introduced by Sz~sz [7--9] and Faith [3]. Later Bass [2] considered perfect rings, i.e. MHR-rings with unit. Let us recall that a ring A is left T-nilpotent if for any sequence of elements a~ of A, i=1, 2, ..., n, ... there exists a positive integer n such that a~a~as...a,=O. For the basic notions and results of the radical theory we refer to Andrunakievi~--Rjabuhin [1], Szfisz [10], Wiegandt [11]. We denote by
TL;DR: In this article, it was shown that the class number of any even unimodular lattice of rank > 2 in a quadratic space having an orthonormal basis over an algebraic field is more than one except in the three cases above.
TL;DR: Brownawell-Waldschmidt as discussed by the authors obtained the theorem Theorem (theorem) of the field of complex, rational, and all algebraic numbers by adjoining an arbitrary transcendental number.
Abstract: Let C.Q. and A be, respectively, the fields of complex, rational, and all algebraic numbers; let Z be the ring of integers. Let Q, be an extension of field Q obtained by adjoining an arbitrary transcendental number e e C, and let Q~ be an algebraic extension of finite degree of the field Q1Waldschmidt [i, 2] and Brownawell [3], applying a method of Gel'fond [4, 5], independently obtained the following result: THEOREM (Brownawell-Waldschmidt). Let Xi, X2 e C. as well as Xi, X2 ~ C , be linearly independent over Q. If Ecau e~,~',, e~a*~A , then the extension of field Q , obtained by adjoining the numbers
Dissertation•
01 Jan 1990
TL;DR: Chinburg as discussed by the authors constructed invariants of the extension N/K lying in the locally free classgroup Cl(ZӶ) by defining weak homological equivalences and their projective invariants over any Noetherian ring A. In the second chapter we generalise Frohlich's canonical factorisability from abelian to arbitrary finite groups.
Abstract: Let N/K be a Galois extension of number fields with Galois group T. T.Chinburg has constructed invariants of the extension N/K lying in the locally free classgroup Cl(ZӶ). In the first chapter we generalise this construction by defining weak homological equivalences and their projective invariants over any Noetherian ring A.In the case where A is an order in a semisimple algebra, we obtain for each A-latticeM an effectively computable subgroup Δ(M) of the kernel group D(A). Specialising tothe case A = ZT we relate Δ subgroups to generalised Swan subgroups and we describe a representative of the coset of the Swan subgroup T(ZӶ) containing Chinburg's invariant Ω(N/K, 1) in terms of the projective invariant of a homomorphism. In the second chapter we generalise A. Frohlich's canonical factorisability from abelian to arbitrary finite groups. We obtain a canonical factorisation function - related to the ring of integers O(_N) - which determines a unique coset of Cl(ZӶ) / D(ZӶ) equal to the coset generated by Chinburg's invariant Ω(N/K, 2). Thus we establish "modulo D(ZӶ)" a conjecture of Chinburg