1. CJM 2016 (vol 69 pp. 258)
 Brandes, Julia; Parsell, Scott T.

Simultaneous Additive Equations: Repeated and Differing Degrees
We obtain bounds for the number of variables required to establish
Hasse principles, both for existence of solutions and for asymptotic
formulÃ¦, for systems of additive equations containing forms
of differing degree but also multiple forms of like degree.
Apart from the very general estimates of Schmidt and BrowningHeathBrown,
which give weak results when specialized to the diagonal situation,
this is the first result on such "hybrid" systems. We also obtain
specialised results for systems of quadratic and cubic forms,
where we are able to take advantage of some of the stronger methods
available in that setting. In particular, we achieve essentially
square root cancellation for systems consisting of one cubic
and $r$ quadratic equations.
Keywords:equations in many variables, counting solutions of Diophantine equations, applications of the HardyLittlewood method Categories:11D72, 11D45, 11P55 

2. CJM 2013 (vol 66 pp. 844)
 Kuo, Wentang; Liu, YuRu; Zhao, Xiaomei

Multidimensional Vinogradovtype 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 Vinogradovtype 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 

3. CJM 2010 (vol 63 pp. 38)
 Brüdern, Jörg; Wooley, Trevor D.

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 nontrivial 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 

4. CJM 2005 (vol 57 pp. 298)
 Kumchev, Angel V.

On the WaringGoldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers
We investigate exceptional sets in the WaringGoldbach 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 

5. CJM 2002 (vol 54 pp. 417)
 Wooley, Trevor D.

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 HardyLittlewood 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 

6. CJM 2002 (vol 54 pp. 71)
 Choi, KwokKwong Stephen; Liu, Jianya

Small Prime Solutions of Quadratic Equations
Let $b_1,\dots,b_5$ be nonzero 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 
