Universal octonary diagonal forms over some real quadratic fields

Abstract

Abstract. In this paper, we will prove there are infinitely many integers n such that n 2 − 1 is square-free and Q( √ n 2 − 1) admits universal octonary diagonal quadratic forms. Mathematics Subject Classification (2000). Primary 11E12, Secondary 11E20.

Number fields without n-ary universal quadratic forms

“…Roughly speaking, if K = Q( √ D) has small class number, it contains many integers of small norm, and we will see that this often forces a universal form to have many variables. On the other hand, Kim [Ki2] showed that there are infinitely many real quadratic fields that admit universal quadratic forms in 8 variables; these fields are all of the form Q( √ n 2 + 1) and have in particular very large class number.…”

Universal Quadratic Forms and Elements of Small Norm in Real Quadratic Fields

“…It may seem that the number of variables required by a universal quadratic form should grow with the discriminant of the (real quadratic) number field. This is not entirely true, as Kim [7] constructed an infinite family of fields of the form Q( √ n 2 − 1) admitting positive diagonal octonary universal forms.…”

The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields

“…All positive definite ternary universal forms over real quadratic fields were determined in [3]. Further developments on universal forms over totally real number fields were established by B. M. Kim (see [12], [13] and [14]). …”

“…Recently, Bhargava and Hanke enunciated that they proved the 290-conjecture which characterizes the universality of (nonclassical) quadratic forms. That is, if a (nonclassical) quadratic form represents the 29 numbers, 1, 2, 3, 5,6,7,10,13,14,15,17,19,21,22,23,26,29,30,31,34,35,37,42,58,93,110,145,203,290, then it is universal (see [9]). One can use this theorem to prove the universality of a Hermitian lattice.…”