Abstract
Let denote the ring of integers of a quadratic field . In 2022, Murtuza and Garge [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022.] gave a necessary and sufficient condition for a diagonal quadratic form where for for representing all matrices over . Let K denote a quadratic field such that its ring of integers is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that is the only imaginary quadratic field with the above properties. Let denote the discriminant of K. We have if and only if 2 is a product of two distinct primes in . With as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form where , to represent all matrices over . This result is a conjecture stated in [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022].
Acknowledgements
We thank Professor S. A. Katre for informing us about the paper of Jungin Lee. The first author would like to thank Dr. Mukesh D. Patil and Dr. Arpita Palchowdhury for support and encouragement. The authors take the opportunity to thank Professor R. M. Pawale, Head, Department of Mathematics, University of Mumbai.
Disclosure statement
No potential conflict of interest was reported by the author(s).