Universal quadratic forms

Gonii: Universal Quaternary Quadratic Forms

We continue our study of quadratic forms using Geometry of Numbers methods by considering universal quaternary positive definite integral forms of square discriminant. We give a small multiple theorem for such forms and use it to prove universality for all nine universal diagonal forms. The most interesting case is x2 + 2y2 + 5z2 + 10w2, which required computer calculations.

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

2-universal Positive Definite Integral Quinary Diagonal Quadratic Forms

As a generalization of the famous four square theorem of Lagrange, Ramanujan found all positive definite integral quaternary diagonal quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary diagonal quadratic forms that represent all positive definite integral binary quadratic forms. §

2-universal Positive Definite Integral Quinary Quadratic Forms

As a generalization of the famous four square theorem of Lagrange, Ramanujan and Willerding found all positive definite integral quaternary quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary quadratic forms that represent all positive definite integral binary quadratic forms. We also discuss recent results on positive definite int...

Representation of Quadratic Forms by Integral Quadratic Forms