The Hermite constant and its generalizations - Oh, Chun Rong (Nanyang Technological University, 2023)
The majority of the values for the Hermite constant and its generalizations, namely the dual Hermite constant, the Rankin constant and the Bergé-Martinet constant, have yet to be found. This thesis initiates an exploration into the missing values for these constants. A novel contribution is made by establishing a lower bound for the Rankin constant γn,2 with the lattice Dn. Using this lower bound, we have narrowed down the list of interesting lattices in dimensions 5, 7, and 9 with potential to achieve γn,2 for n = 5, 7, 9. Conjectures are presented for the values of γ5,2, γ7,2, and γ9,2. Future endeavors could concentrate on proving these conjectures.