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1. CJM 2006 (vol 58 pp. 476)
Apolar Schemes of Algebraic Forms 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.}
Keywords:Waring's problem, apolarity, polar polyhedron Categories:14N05, 14N15 |
2. 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 |