Perfection and Eutaxy
The aim of this chapter is to give a characterization of extreme lattices, those lattices on which the Hermite invariant attains a local maximum. To this end, we introduce in Section 3.2 the two notions of perfection and eutaxy, and prove in Section 3.4 that they characterize extreme lattices; in particular, we obtain an algorithm which allows us to decide whether a given lattice is extreme, whose justification relies on the technical material in Sections 3.1 and 3.3. Then we give in Sections 3.5 and 3.6 a detailed analysis of the properties of perfection and eutaxy.
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