This is a note on the classical Waring's problem for algebraic forms. Fix integers $(n,d,r,s)$, and let $\Lambda $ be a general $r$-dimensional subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let $\mathcal{A}$ denote the variety of $s$-sided polar polyhedra of $\Lambda $. We carry out a case-by-case study of the structure of $\mathcal{A}$ for several specific values of $(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is shown to be a rational variety. In the second batch, $\mathcal{A}$ is a finite set of which we calculate the cardinality.

Estimates are established for the number of integers of size N, in intervals of size $N^\theta$, that fail to admit a representation as the sum of s cubes (s = 5, 6). Thereby it is shown that almost all such integers are represented in the proposed manner. When s = 5 one may take $\theta = 10/21$, and when s = 6 one may take any $\theta >17/63$. Similar such conclusions are also established for the related problem associated with the expected asymptotic formula.

An asymptotic formula is established for the number of representations of a large integer as the sum of kth powers of natural numbers, in which each representation is counted with a homogeneous weight that de-emphasises the large solutions. Such an asymptotic formula necessarily fails when this weight is excessively light.

Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N)$ denote the number of positive integers not exceeding N that are congruent to 4 modulo 24, yet cannot be written as the sum of four squares of prime numbers. A method is described that shows that for each positive number $\epsilon$, one has $E(N) \ll N^{13/30 + \epsilon}$, thereby exploiting effectively the 'excess' fourth square of a prime so as to improve the recent bound $E(N) \ll N^{13/15 + \epsilon}$ due to J. Liu and M.-C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares.

2000 Mathematical Subject Classification: 11P32, 11P05, 11P55.

Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy--Littlewood dissection, these estimates are applied to derive an upper bound for the $s$th moment of the smooth cubic Weyl sum of the expected order of magnitude as soon as $s\ge 7.691$. Related arguments demonstrate that all large integers $n$ are represented as the sum of eight cubes of natural numbers, all of whose prime divisors are at most $\exp (c(\log n\log \log n)^{1/2})$, for a suitable positive number $c$. This conclusion improves a previous result of G. Harcos in which nine cubes are required. 1991 Mathematics Subject Classification: 11P05, 11L15, 11P55.

Let Rk(n) denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R3(n) [Lt ] n5/9+ε, and that R4(n) [Lt ] n47/90+ε, where ε > 0 is arbitrary. This extends work of Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.