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Search: MSC category 11D72 ( Equations in many variables [See also 11P55] )

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1. CJM Online first

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 formulæ, 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 Browning--Heath-Brown, 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 Hardy-Littlewood method
Categories:11D72, 11D45, 11P55

2. CJM 2011 (vol 63 pp. 481)

Baragar, Arthur
The Ample Cone for a K3 Surface
In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$.

Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics
Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05

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 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

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