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1. CMB 2011 (vol 54 pp. 645)
An Extension of Craig's Family of Lattices Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of
unity. The lattices in Craig's family are $(p-1)$-dimensional and
are geometrical representations of the integral $\mathbb
Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a
positive integer. This lattice construction technique is a powerful
one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$,
Craig's lattices are the densest packings known. Motivated by this,
we construct $(p-1)(q-1)$-dimensional lattices from the integral
$\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle
1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$
and $j$ are positive integers. In terms of sphere-packing density,
the new lattices and those in Craig's family have the same
asymptotic behavior. In conclusion, Craig's family is greatly
extended while preserving its sphere-packing properties.
Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields Categories:11H31, 11H55, 11H50, 11R18, 11R04 |
2. CMB 2009 (vol 52 pp. 63)
Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes We prove a new upper bound for the smallest zero $\mathbf{x}$
of a quadratic form over a number field with the additional
restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed
hyperplanes. Our bound is polynomial in the height of the quadratic
form, with an exponent depending only on the number of variables but
not on $m$.
Categories:11D09, 11E12, 11H46, 11H55 |