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

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 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.

On a quadratic Waring's problem with congruence conditions

“…Note that our result presents the same growth as the best known upper bound on g Z (n). More precisely, the upper bound on g ∆ (n) we obtain is approximately n 2 times the upper bound on g Z (n) obtained in [1]. We will adopt geometric language of quadratic spaces, lattices and Z-cosets in studying g ∆ (n) so that we shall use the geometric theory of those.…”

“…Let K be a positive definite Z-lattices of rank n and let {d 1 , ..., d n } be a basis for K. We say that a basis {d 1 , ..., d n } for K is balanced HKZ-reduced if its corresponding Gram matrix M is of the form H[X] := X t HX, where X = (x ij ) ∈ U (n) and H = diag(h 1 , ..., h n ) satisfy the following two properties: 1−x . Note that every positive definite Z-lattice has a "balanced HKZ-reduced" basis (see [1,Section 4]). On the other hand, we can bound the values α(j − i) ([17, Corollary 2.5]) as…”

“…Proof. The proof of this proposition is motivated by Section 6 of [1] and a modification of the arguments in there. The strategy of the proof is outlined as follows.…”

“…Later, Kim-Oh [8] proved that g Z (n) = O(3 n/2 n log n), which improves Icaza's bound. Recently, Beli-Chan-Icaza-Liu [1] obtained a better upper bound g Z (n) = O(e (4+2 √ 2+ε)…”

$g$-invariant on unary Hermitian lattices over imaginary quadratic fields with class number $2$ or $3$

“…The answer is unfortunately no: we will see later that a positive definite unimodular Hermitian lattice is represented by some if and only if is a principle ideal. So, we will restrict our attention to a smaller set as done in [1] and [9]. Denote by (1) the set consisting of all positive definite integral unary Hermitian lattices over O that can be represented by some , and by (1) the smallest positive integer such that every Hermitian lattice in (1) is represented by (1) uniformly.…”