The birational composition of arbitrary quadratic form with binary quadratic form
Abstract
Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n respectively over a field K, charK ≠ 2. Herein, the problem of the birational composition of f(X) and g(Y) is considered, namely, the condition is established when the product f(X)g(Y) is birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The main result of this paper is the complete solution of the problem of the birational composition for quadratic forms f(X) and g(Y) over a field K when m = 2. The sufficient and necessary conditions for the existence of birational composition h(Z) for quadratic forms f(X) and g(Y) over a field K for m = 2 are obtained. The set of quadratic forms is described which can be considered as h(Z) in this case.
References
- Hurwitz A. Über die Komposition der quadratischen Formen. Mathematische Annalen. 1922;88(1–2):1–25. DOI: 10.1007/BF01448439.
- Radon J. Lineare Scharen orthogonaler Matrizen. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 1922;1(1):1–14. DOI: 10.1007/BF02940576.
- Lam KY. Topological methods for studying the composition of quadratic forms. In: Riehm CR, Hambleton I, editors. Quadratic and Hermitian forms [Conference on quadratic forms and Hermitian K-theory, held at McMaster University; 1983 July 11–22; Hamilton, Ontario, Canada]. Providence: American Mathematical Society; 1984. p. 173–192 (CMS conference proceedings; volume 4).
- Pfister A. Multiplikative quadratische Formen. Archiv der Mathematic. 1965;16(1):363–370. DOI: 10.1007/BF01220043.
- Bondarenko AA. [On the birational composition of quadratic forms]. Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series. 2007;4:56–61. Russian.
- Bondarenko AA. [Birational composition of quadratic forms over a local field]. Matematicheskie zametki. 2009;85(5):661–670. DOI: 10.4213/mzm4673. Russian.
- Bondarenko AA. [The birational composition of quadratic forms over a finite field]. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 2010;3:90–93. Russian.
- Bondarenko AA. [The birational composition of quadratic forms over a function field]. Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series. 2014;3:28–32. Russian.
- Bondarenko AA. [The birational composition of ternary quadratic forms]. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 2012;2:106–110. Russian.
- Serre J-P. Cours d’arithmétique. Paris: Presses Universitaires de France; 1970. 188 p. Russian edition: Serre J-P. Kurs arifmetiki. Skopin AI, translator; Malyshev AV, editor. Moscow: Mir; 1972. 184 p.
- Knebusch M, Scharlau W. Algebraic theory of quadratic forms. Generic methods and Pfister forms. Boston: Birkhäuser; 1980. 44 p. (DMV seminar; 1).
Copyright (c) 2022 Journal of the Belarusian State University. Mathematics and Informatics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
The authors who are published in this journal agree to the following:
- The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
- The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
- The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)
