ON THE (NON-)EQUIVALENCE OF INTEGRAL BINARY QUADRATIC FORMS AND THEIR NEGATIVE FORMS

English

Barthel, Jim Jean-Pierre[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC) >]

Müller, Volker[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC) >]

Undated

No

[en] Indefinite integral binary quadratic forms ; equivalence classes ; negative forms

[en] Integral binary quadratic forms have been extensively studied in order to compute the class number of real and complex quadratic ﬁelds. Many studies restricted the equivalence class of quadratic forms or identiﬁed speciﬁc forms in order to compute the class number through enumeration of equivalence classes. One often used assumption is that a given form and its negative are equivalent or are synthetically identiﬁed. This document presents a concise Lagrangian approach to integral binary quadratic forms, outlines the equivalence classes in its broadest sense and uses it to develop a general condition when a given integral binary quadratic form and its negative are equivalent. Then, the existence of integral binary quadratic forms which are non-equivalent to their negatives is proven and an elementary algorithm to determine the (non-)equivalence of a given form and its negative is outlined.