xiv OVERVIEW
on all local optima. Although the lattice covering and the lattice packing-covering
problem have both not been solved in dimension 6 so far, we describe conjecturally
optimal lattices obtained by heuristic methods (see Section 5.3.3). Both lattices
have the property that their Delone triangulation refines the Delone subdivision of
the lattice E6. As a first step in direction of a proof for global optimality, we prove
computationally by a branch-and-bound method that the conjectural optimal cov-
ering lattice is the unique optimum among lattices having this refinement property
(see Theorem 5.28). In many cases it is a problem to obtain exact coordinates for
the new lattices. Exemplarily we show how to obtain exact coordinates for the
new conjectural optimal 6-dimensional packing-covering lattice. We describe a new
conjecturally optimal 7-dimensional covering lattice whose corresponding positive
definite quadratic forms are (to our surprise) even rational (see Section 5.3.5). We
describe how we obtained new, currently best known covering lattices in dimensions
d 9. These computations are based on our new T -secondary cone theory.
In Section 5.4 we apply the theory of T -secondary cones to a problem in al-
gebraic number theory: the classification of all totally real thin algebraic number
fields. Our classification is based on a list of 17 candidates, previously given by
Bayer-Fluckiger and Nebe. By proving lower bounds on the covering density of
positive definite quadratic forms in an associated linear subspaces T we exclude
three of the candidates. This finishes the classification.
In Section 5.5 we take a closer look at two of the most exceptional lattices (in
low dimensions): The root lattice E8 and the Leech lattice. We show that both
lattices are rigid, meaning they are uniquely determined by the type of their Delone
subdivision. By applying our local lower bounds we show that the Leech lattice
is a local lattice covering and packing-covering optimum. This shows in particular
the existence of rigid, locally optimal lattice coverings, which affirmatively answers
a long standing open question of Dickson (1968). The same method can not be
applied to the E8 root lattice. In fact, a local analysis reveals that it is not even a
locally optimal covering lattice. In connection with a proof of this fact, we find a
new currently best known 8-dimensional covering lattice (see Section 5.5.6).
In Section 5.6 we continue the investigation of lattices similar to E8, revealing a
previously unknown phenomenon: The existence of local lattice covering maxima,
for which we derive necessary and sufficient conditions. Based on them we prove
computationally that the E6 root lattice is the only lattice covering maximum in
dimensions less or equal to 6, aside of Z (which trivially is at the same time a
1-dimensional lattice covering optimum). The E8 root lattice itself turns out to be
only “almost” a local lattice covering maximum: Almost any, but not every local
change of an associated positive definite quadratic form yields a lower covering
density. We baptize these lattices covering pessima and prove that a lattice is a
covering pessimum, if the Delone polytopes attaining its covering radius are all
regular cross polytopes. This is for example the case for the root lattices D4 and
E8.
In Section 5.7 we consider local lattice covering maxima within a linear subspace
T in the space of (associated) quadratic forms. We show a connection to the famous
Minkowski conjecture, respectively to the so-called covering conjecture. Based on
our new T -secondary cone theory we obtain an algorithm which can be used to
verify or falsify the covering conjecture in a given dimension. By a recent result of
Curtis McMullen the covering conjecture implies the Minkowski conjecture. Thus
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