Showing papers on "Ring of integers published in 1987"

Polynomials with Low Height and Prescribed Vanishing

Abstract: In a recent paper [2] we obtained an improved formulation of Siegel’s classical result([9],Bd. I,p. 213, Hilfssatz) on small solutions of systems of linear equations. Our purpose here is to illustrate the use of this new version of Siegel’s lemma in the problem of constructing a simple type of auxiliary polynomial. More precisely, let k be an algebraic number field, O k its ring of integers, α1,α2,…,αJ distinct, nonzero algebraic numbers (which are not necesarily in k), and m1,m2,…,mJ positive integers. We will be interested in determining nontrivial polynomials P(X) in 0 K [X] which have degree less than N, vanish at each αj with multiplicity at least mj and have low height. In particular, the height of such plynomials will be bounded from above by a simple function of the degrees and heights of the algebraic numbers αj and the remaining data in the problem: m1,m2,…mJ, N and the field constants associated with k.

Integer parts of powers of quadratic units

Abstract: Let a > 1 be a unit in a quadratic field. The integer part of an, denoted [an], is shown to be composite infinitely often. Provided a $& (1 + V5)/2, it is shown that the number of primes among [a], [.2],..., [an] iS bounded by a function asymptotic to c log2 n, with c = 1/(2 log 2 . log 3). Let a > 1 be a unit in a quadratic field Q(v\IU), with D > 1 a square-free rational integer. It is known in some cases that the integer parts [aXn] of powers of a (n = 1, 2, 3,... ) are composite infinitely often [1]. We show this in general, the proof guaranteeing in fact that infinitely many of the [an] are divisible by [a]. (There is one exceptional case a = (1 + \'-)/2 wherein [a] = 1; here infinitely many of the [aXn] are divisible by [a 2] > 1.) Define f,(x) to mean the number of n, 1 O, s > O. HEURISTIC REMARK. As x -x oc the function 1 + B(x) is asymptotic to c log2 x, where c = 1/(2 log 2 log 3). If one says "m is prime with probability 1/ log m," then [an] is prime with probability about 1/n log a. Summing this for n 1 is a unit of Q(v\U) with D > 1 squarefree. Write tn for [an], and let N(/3) denote the norm and /3' the conjugate of/3 for /3 any integer of Q(-\I5). Then: (a) If N(a) = 1, then tn = (n + -n)1. (b) If N(a) = -1, then | (c>n+X-nn i n is odd, tn(cen + X-n) -1, if n is even. Received by the editors April 15, 1986 and, in revised form, August 18, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 1OH20, 12A25.

The Conner-Floyd Map for Formal A-Modules

Abstract: A generalization of the Conner-Floyd map from complex cobordism to complex /C-theory is constructed for formal A-modules when A is the ring of algebraic integers in a number field or its p-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal A-modules and to confirm a conjecture of D. Ravenel. 0. Introduction. Let BP be the spectrum representing Brown-Peterson cohomology with respect to a prime p and let E be the Adams summand of complex if-theory with respect to this prime. The BP version of the Conner-Floyd map is a map of spectra BP —+ E which induces a natural equivalence BP, X ®Bp. E» ~ E.X. In particular this induces an isomorphism £* ®Bp. BP»BP®bp.£. ~ E.E and so provides a way of computing the Hopf algebra of stable co-operations for E from those for BP. Using this one can obtain a description of E*E similar to that for K*K contained in [AHS]. The study of BP and the computation of BP* BP are based on a study of formal group law, in particular the p-typical formal group law. In [Rl] Ravenel studied a generalization of this situation where the formal group law is replaced by a formal yl-module where A is the ring of integers in an algebraic number field K or its p-adic completion. The purpose of the present paper is to describe the corresponding generalization of the map (BP», BP* BP) —► (E*,E*E) induced by the Conner-Floyd map, and to compute the generalization of E*E. This is of interest because it provides some information about a conjecture (3.10) made in [Rl]. This conjecture concerned the value of a certain Ext group Exty,t(Va, Va) when K is an extension of the field Qp of p-adic numbers. Here (Va, VaT) is the Hopf algebroid corresponding to the A-typical formal A module. This group was conjectured to be, up to small factor, A/J^,^. Here J^iq_1\ is the ideal of A generated by the elements of the form aTM — 1 for units a of A congruent to 1 mod(7r) and (it) is the unique prime ideal in A. We will show, using the generalization of the Conner-Floyd map, that A/J*,^ occurs as E\' in the chromatic spectra sequence for formal ^-modules [Rl, Lemma 2.10] and that the small factor in the conjecture is contributed by the nontriviality of the differential d\ originating from this group. We will analyze this differential and show that it is nonzero for A Received by the editors June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 14L05. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Systems of linear equations with dense univariate polynomial coefficients

Abstract: An algorithm for computing the power series solution of a system of linear equations with components that are dense univariate polynomials over a field is described and analyzed. A method for converting the power series solution to rational form is derived. Theoretical and experimental cost estimates are obtained and used to identify classes of problems for which the power series method outperforms modular methods. Finally, it is shown that the power series method also provides an effective mechanism for solving the problem in which the coefficients of the polynomials are from the ring of integers.

On the Stickelberger ideal and the relative class number

Abstract: Let k be any imaginary abelian fieldX R the integral group ring of G = Gal(k/(2) and S the Stickelberger ideal of k. Roughly speakingX the relative class number hof k is expressed as the index of S in a certain ideal A of R described by means of G and the complex conjugation of k; c-h= [A: S] with a rational number cin -NJ = {n/2;n e NJ} which can be described without hand is of lower than hif the conductor of k is sufficiently large (cf. [6 9 1O]; see also [5]). We shall prove that 2ca natural numberX divides 2([k: (2]/2)lk 1/2. In particularX if k varies through a sequence of imaginary abelian fields of degrees boundedX then ctakes only a finite number of values. On the other handX it will be shown that ccan take any value in 2NJ when k ranges over all imaginary abelian fields. In this connectionX we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields. Let 2, C;!!, 1R, and C denote the rational integer ring, the rational number field, the real number field, and the complex number field, respectively. A finite abelian extension over C;!! contained in C will be called an abelian field. Let k be an imagi- nary abelian field, namely, an abelian field not contained in 1R. We denote by R(k) the group ring of the Galois group G = Gal(k/C;!!) over z and by s(H), for any subgroup H of G, the sum in R(k) of all elements in H. Put A(k) = { E R(k); (1 + jk)°l = as(G) for some a E E}, where ik denotes the complex conjugation of k. Let hk denote the relative class number of k (i.e., the so-called first factor of the class number of k), Qk the unit index of k, 9k the number of distinct prime numbers ramified in k, and S(k) the Stickelberger ideal of k in the sense of Iwasawa-Sinnott, which is an additive sub- group of A(k) with finite index (for the definition of the Stickelberger ideal, see [6, 10]). We define ck as the ratio of the index [A(k): S(k)] to hk: Ck hk = [A(k): S(k)] The product QkCk is known to be a natural number and is determined by Sinnott in various cases, for example, in the case 9k = 1 or 2 (cf. [10]). He has also shown in [9] that, if k is a cyclotomic field, then Ck = 2b where b = 0 or 29k-1-1 according as 9k = 1 or 9k > 2 (for the case 9k = 1, see [6]). In this paper, we shall give an additional result concerning the range of Ck . Received by the editors July 31 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary llR20 llR29; Secondary llN25 llR18.

Non-Vanishing of Certain Values of L -Functions

Abstract: Let K be an imaginary quadratic field. The L-functions that we will consider are defined by $$L(\chi ,{\rm{s}}){\rm{ = }}\sum\limits_a {\frac{{x\left( a \right)}}{{N{{\left( a \right)}^s}}}}$$ where the sum is over the nonzero ideals of the ring of integers OK of K. Here χ is a grossencharacter of K of type Ao. That is, χ is a complex-valued multiplicative function on the ideals of OK such that \(\chi \left( {\left( \alpha \right)} \right){\rm{ = }}{\alpha ^{\rm{n}}}{\bar \alpha ^{\rm{m}}}\) for all α OK, α = 1 (mod fχ), where n, m ∈ Z and fχ is an ideal of OK (the conductor of χ). We call (n,m) the infinity type of χ. The above series defines an analytic function for Re(s) sufficiently large which can be analytically continued to the entire complex plane and satisfies a functional equation. By translating s or applying complex conjugation, we can clearly assume that χ has infinity type (n,0) with n = nχ > 0, as we will from here on. The functional equation is then as follows.

The Near-Ring of Some One Dimensional Noncommutative Formal Group Laws

Abstract: Publisher Summary This chapter discusses the near-ring of some one-dimensional non-commutative formal group laws. In the chapter, A will denote either the ring of integers modulo p 2 , p a prime, or K[t]/(t 2 ), where K is a field of characteristic p, and t is an indeterminant.

On the $p$-adic heights of some abelian varieties

Abstract: For an abelian variety defined over an algebraic number field, different definitions of p-adic heights have been given by several authors. In this note, we shall prove that the p-adic height defined by A. Neron and that by P. Schneider coincide. 1. In this section we shall recall briefly the construction of p-adic heights. Let K be a finite extension field of the p-adic rational number field Qp, where p is a prime number. Let 0 be the integer ring of K, p the maximal ideal of 0, and let k = 0/P be the residue field of 0. Let A be an abelian variety of dimension d defined over K, 6 = idA the identity map of A. Let 0 be the identity point of A and let t1, ... , td be a p-admissible system of local coordinates of A at 0 in the sense of A. Neron; namely, t1,.. ., td are K-rational functions on A which constitute a local coordinate system at 0 and their reductions modulo p are k-rational functions on A x o k also constitute a local coordinate system at the identity point of A x o k. Let A(K) be the group of K-rational points of A and let U be an open neighborhood of 0 in A(K). A function 4>: U -+ K is called 0-analytic on U if for a c U we have (D(a) = 0(t 1 (a), * ,td (a)), where X = (Tl,..., Td) is a d-variable power series with coefficients in 0 which converges at (ti(a), ... , td(a)) for all a c U. A K-rational divisor A on A is called disjoint from 0 mod p if any component of the set obtained from the reduction mod p of the support of A does not contain the identity point of A x o k. For a subgroup G of A(K), let A = Z[G] be the group ring of G with coefficients in Z, which is also the group of 0-cycles with components in G. Let I c A be the augmentation ideal of A; i.e., the ideal generated by the cycles of the form (a) (0) with a E G. Let 12 C A be the ideal generated by the cycles of the form (a + b)-(a)-(b) + (0) with a, b C G. As (a + b)-(a)-(b) + (0) = ((a) (0)) * ((b) (0)), where * is the multiplication in A, 12 coincides with the square of I; I2 {a * bla, b e I}. A divisor and a 0-cycle are called disjoint if their supports are disjoint. For a divisor A = div(f) linearly equivalent to 0, and a 0-cycle a E I which are disjoint, we define a pairing [A,a] by [A,a] = Hlf(ai)mi for a = Emi(ai), as usual. For a divisor A algebraically equivalent to 0 and a 0-cycle a = E mi(ai), let A * a = E miAai, where Aai is the translate of A by ai, and let a = mlm,(-ai). If a E I, then A * a is linearly equivalent to 0. So for a = b * c EE 2, A algebraically Received by the editors August 7, 1985 and, in revised form, March 21, 1986. 1980 Mathematics S*ject Cassification (1985 Reviin). Primary 14K15. (?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

Algorithm for maximizing a linear function on the set of integral points of a convex polyhedron

Abstract: In this paper we present a new algorithm for solving the following problem of integral linear programming: find the maximum of the linear function fo (x) = clx: qc2x~ -5 "-5 c,~x,~ ( i ) on the finite set of integral points of the convex polyhedron V, defined by the finite system of linear inequalities f~ (x) = ai:xl + ai~x2 + . . . + a~xn <~ hi, (2) ~= 1,m. It is assumed here that aij, b i, cj (i = i, m, j = I, n) belong to the ring of integers Z. In Sec. i the algorithm of [i] for finding the set of integral points of the polyhedron V is generalized; in Sec. 2 an algorithm for solving the problem of integral linear programming for the case when the set of its admissible solutions is bounded is described.