Universality of certain diagonal quadratic forms for matrices over a ring of integers

Murtuza Nullwala () and Anuradha S. Garge ()
Additional contact information
Murtuza Nullwala: Ramrao Adik Institute of Technology
Anuradha S. Garge: University of Mumbai

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 1, 54-68

Abstract: Abstract In 2018, Jungin Lee [5] gave a necessary and sufficient condition for a diagonal quadratic form $$\sum _{i=1}^{m}a_iX_i^2$$ ∑ i = 1 m a i X i 2 where $$a_i\in {\mathbb {Z}}$$ a i ∈ Z for all i, $$1\le i \le m$$ 1 ≤ i ≤ m for representing all $$2\times 2$$ 2 × 2 matrices over $${\mathbb {Z}}$$ Z . In this paper, we will consider the imaginary quadratic field $${\mathbb {Q}}(\sqrt{-7})$$ Q ( - 7 ) . Its ring of integers $${\mathcal {O}}$$ O is a principal ideal domain. $${\mathbb {Q}}(\sqrt{-7})$$ Q ( - 7 ) is the only imaginary quadratic field such that $${\mathcal {O}}$$ O is a principal ideal domain and 2 is a product of two distinct primes in $${\mathcal {O}}$$ O (upto units). With $${\mathcal {O}}$$ O as above, in this paper we give a necessary and sufficient condition for a diagonal quadratic form $$a_1X_1^2+a_2X_2^2+a_3X_3^2$$ a 1 X 1 2 + a 2 X 2 2 + a 3 X 3 2 where $$a_1,a_2,a_3\in {\mathcal {O}}$$ a 1 , a 2 , a 3 ∈ O to represent all $$2\times 2$$ 2 × 2 matrices over $${\mathcal {O}}$$ O .

Keywords: Imaginary quadratic number field; Ring of integers; Diagonal quadratic form; Universal form; Principal ideal domain; 11A05; 11E25; 15A24; 15B33 (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-022-00345-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:55:y:2024:i:1:d:10.1007_s13226-022-00345-3

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-022-00345-3

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().