Real quadratic fields admitting universal lattices of rank7

TL;DR

This paper proves that for sufficiently large square-free positive integers d, there are no universal quadratic lattices of rank 7 over the ring of integers in the quadratic field Q(√d).

Contribution

It establishes a non-existence result for universal lattices of rank 7 in large quadratic fields, extending understanding of universal quadratic forms.

Findings

No integral septenary universal quadratic lattices over  for large square-free d.

Provides bounds on d for the non-existence of such lattices.

Contributes to the classification of universal quadratic forms over quadratic fields.

Abstract

In this paper, we will prove that if $d$ is sufficiently large square-free positive rational integer, then there is no integral septenary universal quadratic lattice over $\mathcal{O}_F$ where $F=\mathbb Q(\sqrt{d})$.