On quadratic Waring’s problem in totally real number fields

Abstract

We improve the bound of the g g -invariant of the ring of integers of a totally real number field, where the g g -invariant g ( r ) g(r) is the smallest number of squares of linear forms in r r variables that is required to represent all the quadratic forms of rank r r that are representable by the sum of squares. Specifically, we prove that the g O K ( r ) g_{\mathcal {O}_K}(r) of the ring of integers O K \mathcal {O}_K of a totally real number field K K is at most g Z ( [ K : Q ] r ) g_{\mathbb {Z}}([K:\mathbb {Q}]r) . Moreover, it can also be bounded by g O F ( [ K : F ] r + 1 ) g_{\mathcal {O}_F}([K:F]r+1) for any subfield F F of K K . This yields a subexponential upper bound for g ( r ) g(r) of each ring of integers (even if the class number is not 1 1 ). Further, we obtain a more general inequality for the lattice version G ( r ) G(r) of the invariant and apply it to determine the value of G ( 2 ) G(2) for all but one real quadratic field.