Showing papers on "Ring of integers published in 1984"

Parallel computation for well-endowed rings and space-bounded probabilistic machines

Abstract: It is shown that a probabilistic Turing acceptor or transducer running within space bound S can be simulated by a time S2 parallel machine and therefore by a space S2 deterministic machine. (Previous simulations ran in space S6.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic parallel time to computations having small Boolean parallel time, and this development is applied to computing the completion of stochastic matrices. The method introduces a generalization of the ring of integers, called well-endowed rings. Such rings possess a very efficient parallel implementation of the basic (+,−,×) ring operations.

Congruence Properties of Zariski‐Dense Subgroups I

Abstract: This paper deals with the following general situation: we are given an algebraic group G defined over a number field K, and a subgroup F of the group G(K) of K-rational points of G. Then what should it mean for F to be a 'large' subgroup? We might require F to be a lattice in G, to be arithmetic, to contain many elements of a specific kind, to have a large closure in some natural topology, et cetera. There are many theorems proving implications between conditions of'size' of this kind. We shall consider the case of G a K-simple group, usually Q-simple. The Zariski topology of G, for which the closed sets are those defined by the vanishing of polynomial functions, is very coarse. On the other hand, the various valuations v of K each give rise to a 'strong' topology on G(K); if v is non-archimedean, and Kv is the completion of K with respect to v, then G(KV) is a non-archimedean Lie group, and if £>„ is the ring of integers of Kv, the open subgroup G(£5 J of G(KV) of integral points is defined for all but finitely many v. The Zariski closure F of a subgroup F of G(K) is a /C-algebraic subgroup of G, while the strong closure Fy is a Lie subgroup of G(KV). For elementary reasons the dimension of Tv is at most the dimension of F. Our results are in the opposite direction: we show, under suitable conditions on G, that if F = G then for almost all v we have that Tv contains G(OV). To illustrate this with a specific example, if Ml 5. . . , Mk are matrices in SL2(Q) generating a group F which is Zariskidense in SL2, then for all sufficiently large prime numbers p we have F p = SL2(Zp). Further, this result is effective, in the sense that we could, in principle, check the hypothesis for given Mx,..., Mk, and derive a bound on p beyond which the conclusion holds. (It is perhaps worth remarking that this special case, which is the simplest application of our results, may be proved with much less effort than the general theorem.)

Conjugacy classes of hyperbolic matrices in (,) and ideal classes in an order

Abstract: A bijection is proved between Sl(7s,Z?-conjugacy classes of hyperbolic matrices with eigenvalues { A1, . . ., An} which are units in an n-degree number field, and narrow ideal classes of the ring R;, = Z[A,]. A bijection between Gl(n, Z)-conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof. In 1980, Peter Sarnak was able to obtain an estimate of the growth of the class number of real quadratic number fields using the Selberg Trace Formula and a bijection between hyperbolic elements of S1(2, Z) and quadratic forms. The "class number" counted was the number of congruence classes of quadratic forms as studied by Gauss [1]. In this paper we will translate this bijection into modern number-theoretic terms by counting ideal classes in a ring of integers associated to a given field. In this way a bijection is proved between conjugacy classes of hyperbolic matrices in S1(2, Z) with a given set of eigenvalues and ideal classes in a certain order (i.e. subring of dimension n over Z) associated to the ring of integers OK in a real n th degree number field K. This more direct method is necessary for generalizing the bijection to higher dimensional cases because Sarnak's result depends upon quadratic forms, Pell's equation and other things which are well understood only ill the case of S1(2, Z). We must mention the work of Latimer and MacDuffee [3] who first proved Theorem 2 in a slightly different fashion. Important also is the extensive work of Taussky [7-9], who simplified the results of Latimer and MacDuffee and extended them in certain directions, as well as doing much work on the S1(2, Z) case. It follows from a brief examination of the characteristic polynomial for a matrix A in Sl(n, Z) that the eigenvalues of A are conjugate units in an extension of Q. We shall insist in the remainder of this paper that A be "hyperbolic" with irreducible characteristic polynomial, that is, A will have distinct real eigenvalues A('), each of which is of degree n over Q. PROPOSITION 1. If A is an eigenvalue for a matrix A E SL(n, Z), then for any field K containing A there exists an eigenvector w TS = (@1S.. ., Xn), with wi E OK. Received by the editors March 30, 1983. 1980 Mathematics Subject Classification. Primary 10-02, 15-02; Secondary 15A18, 15A36, lOC07. tw1984 American Mathematical Society 0025-5726/84 $1.00 + $.25 per page

Auslander-reiten quivers of schurian orders

Abstract: We shall consider orders ⋀ over a complete discrete rank one valuation ring R in a split full matrix ring containing a complete sex of primitive orthogonal idempot ents. In case s of finite lattice type and R is the power series ring in one variable over its residexe class field k , we give a description of its index composable lattices and its Auslander-Reiten quiver in terms of representations of partially ordered sets. By a model theoretic argument, this implies a description of all indecomposable lattices if R is the completion of the ring of integers in an algebraic number field at all but possibly a finite number of primes.

Multiplication Of Polynomials Over The Ring Of Integers

Abstract: Let R be a ring, and let f(/spl alpha/), g(/spl alph/) /spl epsi/ R[/spl alpha/] be univariate polynomials over R of degree n We Present an algorithm for computing the coefficients of the product f(/spl alpha/)G (/spl alpha/) by O (nlgn) multiplications This algorithm is based on an algorithm for multiplying polynomials over the ring of integers, and does not depend on R Also we prove that multiplying the third degree polynomials over the ripg of integers requires at least nine multiplications This bound is tight

VLSI Architectures for the Multiplication of Integers Modulo a Fermat Number

Abstract: Multiplication is central in the implementation of Fermat number transforms and other residue number algorithms. There is need for a good multiplication algorithm that can be realized easily on a very large scale integration (VLSI) chip. The Leibowitz multiplier is modified to realize multiplication in the ring of integers modulo a Fermat number. This new algorithm requires only a sequence of cyclic shifts and additions. The designs developed for this new multiplier are regular, simple, expandable, and, therefore, suitable for VLSI implementation.

Finite subgroups of formal -modules over -adic integer rings

Abstract: Let B D A be p-adic integer rings such that A/Zp is finite and B/A is unramified. Generalizing a result of Fontaine on finite commutative p-group schemes, we show that galois homomorphisms of finite subgroups of one-dimensional formal A-modules over B are given by power series. Introduction. Let K be a finite extension of the rational p-adic number field Qp, and A the integer ring of K. Let L be a complete unramified extension of K, B the ring of integers of L, and p the maximal ideal of B. We write p for the maximal ideal of the integer ring of the algebraic closure L of L. Let F denote an n-dimensional formal A-module defined over B of finite A-height. Then F induces an A-module structure on n, which we denote by F(p); it is an A[Oj-module, where e = Gal(L/L). Let P be a finite sub-A[5j-module of F(p) (henceforth, simply of F). In this paper, we attach to P a couple ML(P) of modules over a noncommutative power series ring. Let G be another formal A-module over B of finite A-height and Q be a finite sub-A[05j-module of G. Then we describe the A[0jhomomorphisms from P to Q by morphisms from ML(Q) to ML(P) (Theorem 1). If A = Zp (the p-adic integer ring), this result follows from Fontaine [4], but our proof depends rather on Tate modules of formal groups. Furthermore, if F and G are one-dimensional, we can show that every A[05I-homomorphism from P to Q is of the form g-l a ef for some c E B, where f and g are the logarithms of F and G, respectively (Theorem 3). In [81, Lubin has obtained a rather weaker version of this result. In the following, let K, A, L, B, p, p and f5 be as above. We write 7r for a fixed prime element of A and q for the cadinality of the residue field of A. Let a denote the Frobenius automorphism of L/K. We write E = B,[[Tjj for the ring of noncommutative power series ring over B in a variable T with respect to the multiplication rule: Tb b?T for all b E B. Call FA(B) the category of finite-dimensional formal A-modules over B of finite A-height. I would like to thank the referee for calling my attention to Lubin [81. 1. Homomorphisms of finite subgroups of formal A-modules. We write T(F) for the Tate module of a formal A-module F. T(F) is an A[Oj-module, Afree of rank h (= A-height of F). Let DH' be the category defined in Decauwert [2j. Let M(F) and L(F) be as in [2j; M(F) is an E-module, B-free of rank h and Received by the editors February 15, 1983. 1980 Mathematics Subject Classification. Primary 14L05. Key wouds and phrases. Formal module (group), Tate module, special element, logarithm of formal group. (?)1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page 765 This content downloaded from 157.55.39.203 on Thu, 20 Oct 2016 04:18:24 UTC All use subject to http://about.jstor.org/terms