On a Waring’s problem for integral quadratic and Hermitian forms

Abstract

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n 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 ) g_\mathbb {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–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal 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 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.