2 x 2 Integer Matrices: Composition of Binary Quadratic Forms

EasyChair Preprint no. 11692, version 3

Abstract

       In this  research  paper, we  consider  2 x 2  integer  matrices  and  identify  interesting  binary  quadratic  forms  which  naturally  arise.  Specifically,  we  consider such  symmetric  integer  matrices  and  derive  compositions  of  pure  binary  quadratic  forms  naturally  arising  in  association   with  determinant  of  such  matrices.  We  also,  discover  number-theoretic  results  associated  with  trinary  quadratic  forms  naturally  arising  in  connection  with  2 x  2  symmetric  integer  matrices.  We  formulate  a  “generalized  Waring  problem”  using  real  quadratic  algebraic  numbers. We  also  discuss  composition  of  binary quadratic  forms  naturally  arising  in  other  interesting  structured  2 x  2  integer  matrices. We  explore  representation  of   integers  using  trinary  as  well  as  binary  quadratic  forms.

Keyphrases: eigenvalues, Genus Theorem, integer matrices, quadratic forms

@Booklet{EasyChair:11692,
  author = {Rama Garimella},
  title = {2 x  2  Integer  Matrices: Composition  of  Binary  Quadratic  Forms},
  howpublished = {EasyChair Preprint no. 11692},

  year = {EasyChair, 2024}}