RETRACTED ARTICLE: Linear sections of determinantal varieties

Research Article
Open access

RETRACTED ARTICLE: Linear sections of determinantal varieties

Xiaoxi Zhou 1*
  • 1 Nanjing Foreign Language School    
  • *corresponding author xiaoxizhou27@gmail.com
Published on 27 August 2024 | https://doi.org/10.54254/2753-8818/42/20240645
TNS Vol.42
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-495-8
ISBN (Online): 978-1-83558-496-5

Abstract

Denote Mm×n(k) to be the set of all m × n matrices over k, and PMm×n(k) to be its affine space. Let Dm,n,r ⊂ PMm×n(k) be the subvariety consisting of the projective classes of all m × n matrices with rank less than or equal to r. We study the minimal integer k > 0 such that for any linear subspace L of dimension k in PMm×n(k), Dm,n,r ∩ L ̸= ∅. We get some partial results about this problem. Specially, when m = n = r + 1, we solve this problem for k = R, C and Q.

Keywords:

determinantal varieties, Hurwitz–Radon function, linear section, amicable pair

Zhou,X. (2024). RETRACTED ARTICLE: Linear sections of determinantal varieties. Theoretical and Natural Science,42,271-274.
Export citation

References

[1]. J. F. Adams, Vector fields on spheres, Ann. of Math. 75, 603-632, 1962.

[2]. J. F. Adams, P. Lax and R. Phillips, On matrices whose real linear combinations are non-singular, Proc. Amer. Math. Soc., 16:318-322, 1965. Corrections to “On matrices whose real linear combinations are non-singular”, Proc. Amer. Math.Soc., 17: 945-947, 1966.

[3]. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves, Volume I, 1985, Springer.

[4]. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1111-1112, 2000.

[5]. K. Y. Lam, and D. Randall, Geometric dimension of bundles on real projective spaces, In: Homotopy Theory and Its Applications, a conference on algebraic topology in honor of Samuel Gitler, August 9-13, 1993, Cocoyoc, Mexico (A.Adem, R. J. Milgram and D. C. Ravenel, eds.), Contemp. Math. 188, Amer. Math. Soc., Providence, 129–152.

[6]. S. Lang, Algebra, Graduate Texts in Mathematics 211, Springer.

[7]. Z. Z. Petrovi´c, On nonsingular matrices and Bott periodicity, Publications de l’Institut Math´ematique 65(79).85: 97-102, 1999.

[8]. I. R. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, Springer, 2013.

[9]. D. B. Shapiro, Compositions of Quadratic Forms, de Gruyter Expositions in Mathematics 33, 2000.

[10]. H. Weyl, The classical groups, Princeton University Press, 1946.


Cite this article

Zhou,X. (2024). RETRACTED ARTICLE: Linear sections of determinantal varieties. Theoretical and Natural Science,42,271-274.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

Disclaimer/Publisher's Note

The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of EWA Publishing and/or the editor(s). EWA Publishing and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

About volume

Volume title: Proceedings of the 2nd International Conference on Mathematical Physics and Computational Simulation

ISBN:978-1-83558-495-8(Print) / 978-1-83558-496-5(Online)
Editor:Anil Fernando, Gueltoum Bendiab
Conference website: https://www.confmpcs.org/
Conference date: 9 August 2024
Series: Theoretical and Natural Science
Volume number: Vol.42
ISSN:2753-8818(Print) / 2753-8826(Online)

© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. Authors who publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See Open access policy for details).

References

[1]. J. F. Adams, Vector fields on spheres, Ann. of Math. 75, 603-632, 1962.

[2]. J. F. Adams, P. Lax and R. Phillips, On matrices whose real linear combinations are non-singular, Proc. Amer. Math. Soc., 16:318-322, 1965. Corrections to “On matrices whose real linear combinations are non-singular”, Proc. Amer. Math.Soc., 17: 945-947, 1966.

[3]. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves, Volume I, 1985, Springer.

[4]. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1111-1112, 2000.

[5]. K. Y. Lam, and D. Randall, Geometric dimension of bundles on real projective spaces, In: Homotopy Theory and Its Applications, a conference on algebraic topology in honor of Samuel Gitler, August 9-13, 1993, Cocoyoc, Mexico (A.Adem, R. J. Milgram and D. C. Ravenel, eds.), Contemp. Math. 188, Amer. Math. Soc., Providence, 129–152.

[6]. S. Lang, Algebra, Graduate Texts in Mathematics 211, Springer.

[7]. Z. Z. Petrovi´c, On nonsingular matrices and Bott periodicity, Publications de l’Institut Math´ematique 65(79).85: 97-102, 1999.

[8]. I. R. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, Springer, 2013.

[9]. D. B. Shapiro, Compositions of Quadratic Forms, de Gruyter Expositions in Mathematics 33, 2000.

[10]. H. Weyl, The classical groups, Princeton University Press, 1946.