In this paper we consider a generalization to algebraic number fields of the classical Hermite constant γn. For this constant we extend the well‐known Minkowski bound and study the notion of extreme forms in this setting.

For each positive integer n, let g Z (n) be the smallest integer such that if an integral quadratic form in n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z (n) squares of integral linear forms. We show that as n goes to infinity, the growth of g Z (n) is at most an exponential of √ n. Our result improves the best known upper bound on g Z (n) which is in the order of an exponential of n. We also define an analogous number g * O (n) for writing hermitian forms over the ring of integers O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1, we show that the growth of g * O (n) is at most an exponential of √ n. We also improve results of Conway-Sloane [2] and Kim-Oh [14] on s-integral lattices.

No abstract

No abstract

No abstract