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On lattice bases with special properties. (English) Zbl 0972.11060
Let \(L\) be a lattice in \(\mathbb R^n\). A set of vectors \(\{b_1, \ldots, b_n\}\) is called cyclic non-negative if the coordinates of the \(b_i = (b_{i1}, \ldots, b_{ir})\) satisfy the inequalities \(b_{ii} \geq b_{i,i+1} \geq \ldots \geq b_{i,r} \geq b_{i,1} \geq \ldots \geq b_{i,i-1}\). The authors show that each lattice contains a sublattice of finite index possessing a cyclic non-negative basis, and that this index is \(1\) (\(\leq 24\)) for \(r = 2\) (\(r = 3\)). This concept is used for studying fundamental domains with respect to multiplicative equivalence, which in turn is useful for computing Shintani cones in totally real number fields. Applications to the calculation of special values of zeta functions will be discussed in a subsequent paper.

MSC:
11H99 Geometry of numbers
11R16 Cubic and quartic extensions
Software:
Maple; KANT/KASH
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References:
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