226. CMB 1999 (vol 42 pp. 25)
 Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.

On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions
Analogues of van der Waerden's theorem on arithmetic progressions
are considered where the family of all arithmetic progressions,
$\AP$, is replaced by some subfamily of $\AP$. Specifically, we
want to know for which sets $A$, of positive integers, the
following statement holds: for all positive integers $r$ and $k$,
there exists a positive integer $n= w'(k,r)$ such that for every
$r$coloring of $[1,n]$ there exists a monochromatic $k$term
arithmetic progression whose common difference belongs to $A$. We
will call any subset of the positive integers that has the above
property {\em large}. A set having this property for a specific
fixed $r$ will be called {\em $r$large}. We give some necessary
conditions for a set to be large, including the fact that every
large set must contain an infinite number of multiples of each
positive integer. Also, no large set $\{a_{n}: n=1,2,\dots\}$ can
have $\liminf\limits_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} > 1$.
Sufficient conditions for a set to be large are also given. We
show that any set containing $n$cubes for arbitrarily large $n$,
is a large set. Results involving the connection between the
notions of ``large'' and ``2large'' are given. Several open
questions and a conjecture are presented.
Categories:11B25, 05D10 

227. CMB 1999 (vol 42 pp. 78)
 González, Josep

Fermat Jacobians of Prime Degree over Finite Fields
We study the splitting of Fermat Jacobians of prime
degree $\ell$ over an algebraic closure of a finite field of
characteristic $p$ not equal to $\ell$. We prove that their
decomposition is determined by the residue degree of $p$ in the
cyclotomic field of the $\ell$th roots of unity. We provide a
numerical criterion that allows to compute the absolutely simple
subvarieties and their multiplicity in the Fermat Jacobian.
Categories:11G20, 14H40 

228. CMB 1998 (vol 41 pp. 488)
 Sun, Heng

Remarks on certain metaplectic groups
We study metaplectic coverings of the adelized group of a split
connected reductive group $G$ over a number field $F$. Assume its
derived group $G'$ is a simply connected simple Chevalley
group. The purpose is to provide some naturally defined sections
for the coverings with good properties which might be helpful when
we carry some explicit calculations in the theory of automorphic
forms on metaplectic groups. Specifically, we
\begin{enumerate}
\item construct metaplectic coverings of $G({\Bbb A})$ from those
of $G'({\Bbb A})$;
\item for any nonarchimedean place $v$, show the section for a
covering of $G(F_{v})$ constructed from a Steinberg section is an
isomorphism, both algebraically and topologically in an open
subgroup of $G(F_{v})$;
\item define a global section which is a product of local sections
on a maximal torus, a unipotent subgroup and a set of
representatives for the Weyl group.
Categories:20G10, 11F75 

229. CMB 1998 (vol 41 pp. 328)
 Mollin, R. A.

Class number one and primeproducing quadratic polynomials revisited
Over a decade ago, this author produced class number one criteria for
real quadratic fields in terms of primeproducing quadratic
polynomials. The purpose of this article is to revisit the problem
from a new perspective with new criteria. We look at the more general
situation involving arbitrary real quadratic orders rather than the
more restrictive field case, and use the interplay between the various
orders to provide not only more general results, but also simpler proofs.
Categories:11R11, 11R09, 11R29 

230. CMB 1998 (vol 41 pp. 335)
231. CMB 1998 (vol 41 pp. 187)
 Loh, W. K. A.

Exponential sums on reduced residue systems
The aim of this article is to obtain an upper bound for the exponential sums
$\sum e(f(x)/q)$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$
and $e(\alpha)$ denotes $\exp(2\pi i\alpha)$.
We shall show that the upper bound depends only on the values of $q$ and $s$, %% where $s$ is the number of terms in the polynomial $f(x)$.
Category:11L07 

232. CMB 1998 (vol 41 pp. 158)
 Gaál, István

Power integral bases in composits of number fields
In the present paper we consider the problem of finding power
integral bases in number fields which are composits of two
subfields with coprime discriminants. Especially, we consider
imaginary quadratic extensions of totally real cyclic number
fields of prime degree. As an example we solve the index form
equation completely in a two parametric family of fields of degree
$10$ of this type.
Categories:11D57, 11R21 

233. CMB 1998 (vol 41 pp. 71)
 Hurrelbrink, Jurgen; Rehmann, Ulf

Splitting patterns and trace forms
The splitting pattern of a quadratic form $q$ over
a field $k$ consists of all distinct Witt indices that occur for $q$
over extension fields of $k$. In small dimensions, the complete list
of splitting patterns of quadratic forms is known. We show that
{\it all\/} splitting patterns of quadratic forms of dimension at
most nine can be realized by trace forms.
Keywords:Quadratic forms, Witt indices, generic splitting. Category:11E04 

234. CMB 1998 (vol 41 pp. 86)
235. CMB 1998 (vol 41 pp. 125)
 Boyd, David W.

Uniform approximation to Mahler's measure in several variables
If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure
of $f$, $M(f)$ is defined to be the geometric mean of $f$ over the $k$torus
$\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy
$d2^{n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove
that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials
of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number
of arithmetic operations from the coefficients of $f$ this also demonstrates
an effective (but impractical) method for computing $M(f)$ to arbitrary
accuracy.
Categories:11R06, 11K16, 11Y99 

236. CMB 1998 (vol 41 pp. 15)
 Brown, Tom; Shiue, Peter JauShyong; Yu, X. Y.

Sequences with translates containing many primes
Garrison [3], Forman [2], and Abel and Siebert [1] showed that for all positive integers
$k$ and $N$, there exists a positive integer $\lambda$ such that $n^k+\lambda$ is
prime for at least $N$ positive integers $n$. In other words, there exists $\lambda$
such that $n^k+\lambda$ represents at least $N$ primes.
We give a quantitative version of this result. We show that there exists
$\lambda \leq x^k$ such that $n^k+\lambda$, $1\leq n\leq x$, represents at
least $(\frac 1k+o(1)) \pi(x)$ primes, as $x\rightarrow \infty$. We also give some
related results.
Category:11A48 
