5 papers
No proper generalized quadratic forms are universal over quadratic fields
Ondřej Chwiedziuk, Matěj Doležálek, Simona Hlavinková +5
We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be…
Generalizing Hurwitz's quaternionic proof of Lagrange's and Jacobi's four-square theorems
Matěj Doležálek
A proof of Lagrange's and Jacobi's four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orde…
Representing rational integers by generalized quadratic forms over quadratic fields
Ondřej Chwiedziuk, Matěj Doležálek, Emma Pěchoučková +5
We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive defin…
The Six-Vertex Model with a Non-Standard Boundary Condition
Matěj Doležálek, Martin Raška, Ester Sgallová +2
We consider the enumeration of states in the Brubaker-Bump-Friedberg six-vertex model, whose boundary conditions are determined by an integer partition. In general, we find the num…
Quaternions and universal quadratic forms over number fields
Matěj Doležálek
We study quadratic forms over totally real number fields by using an associated ring of quaternions. We examine some properties of residue class rings of these quaternions and use…