Abstract

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 every positive definite integral quadratic form is equivalent to what we call a balanced Hermite-Korkin-Zolotarevreduced form and use it to show that the growth of g(Z)(n) is at most an exponential of root n. Our result improves the best known upper bound on g(Z)(n) which is on 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 root n. We also improve on results of both Conway and Sloane and Kim and Oh on s-integrable lattices.