Quaternary universal forms over $\mathbf{Q}(\sqrt{13})$

Abstract

Let \(F=\mathbf{Q}(\sqrt{m})\) be a real quadratic field over Q with m a square-free positive rational integer and \(\mathcal{O}\) be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x1,…,xn)=∑1≤i,j≤nαijxixj ( \(\alpha_{ij}=\alpha_{ji}\in \mathcal{O}\) ) is called universal if f represents all totally positive integers in \(\mathcal{O}\) . Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12.