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1. CJM 2013 (vol 66 pp. 844)
Multidimensional Vinogradov-type Estimates in Function Fields Let $\mathbb{F}_q[t]$ denote the polynomial ring over the finite
field $\mathbb{F}_q$.
We employ Wooley's new efficient congruencing method to prove
certain multidimensional Vinogradov-type estimates in $\mathbb{F}_q[t]$.
These results allow us to apply a variant of the circle method
to obtain asymptotic formulas for a system connected to the problem
about linear spaces lying on hypersurfaces defined over $\mathbb{F}_q[t]$.
Keywords:Vinogradov's mean value theorem, function fields, circle method Categories:11D45, 11P55, 11T55 |
2. CJM 2010 (vol 63 pp. 38)
Asymptotic Formulae for Pairs of Diagonal Cubic Equations
We investigate the number of integral solutions possessed by a
pair of diagonal cubic equations in a large box. Provided that the number of
variables in the system is at least fourteen, and in addition the number of
variables in any non-trivial linear combination of the underlying forms is at
least eight, we obtain an asymptotic formula for the number of integral
solutions consistent with the product of local densities associated with the
system.
Keywords:exponential sums, Diophantine equations Categories:11D72, 11P55 |
3. CJM 2005 (vol 57 pp. 298)
On the Waring--Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers We investigate exceptional sets in the Waring--Goldbach problem. For
example, in the cubic case, we show that all but
$O(N^{79/84+\epsilon})$ integers subject to the necessary local
conditions can be represented as the sum of five cubes of primes.
Furthermore, we develop a new device that leads easily to similar
estimates for exceptional sets for sums of fourth and higher powers of
primes.
Categories:11P32, 11L15, 11L20, 11N36, 11P55 |
4. CJM 2002 (vol 54 pp. 417)
Slim Exceptional Sets for Sums of Cubes We investigate exceptional sets associated with various additive
problems involving sums of cubes. By developing a method wherein an
exponential sum over the set of exceptions is employed explicitly
within the Hardy-Littlewood method, we are better able to exploit
excess variables. By way of illustration, we show that the number of
odd integers not divisible by $9$, and not exceeding $X$, that fail to
have a representation as the sum of $7$ cubes of prime numbers, is
$O(X^{23/36+\eps})$. For sums of eight cubes of prime numbers, the
corresponding number of exceptional integers is $O(X^{11/36+\eps})$.
Keywords:Waring's problem, exceptional sets Categories:11P32, 11P05, 11P55 |
5. CJM 2002 (vol 54 pp. 71)
Small Prime Solutions of Quadratic Equations Let $b_1,\dots,b_5$ be non-zero integers and $n$ any integer. Suppose
that $b_1 + \cdots + b_5 \equiv n \pmod{24}$ and $(b_i,b_j) = 1$ for
$1 \leq i < j \leq 5$. In this paper we prove that
\begin{enumerate}[(ii)]
\item[(i)] if $b_j$ are not all of the same sign, then the above
quadratic equation has prime solutions satisfying $p_j \ll \sqrt{|n|}
+ \max \{|b_j|\}^{20+\ve}$; and
\item[(ii)] if all $b_j$ are positive and $n \gg \max \{|b_j|\}^{41+
\ve}$, then the quadratic equation $b_1 p_1^2 + \cdots + b_5 p_5^2 =
n$ is soluble in primes $p_j$.
\end{enumerate}
Categories:11P32, 11P05, 11P55 |