RETRACTED ARTICLE: Linear sections of determinantal varieties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RETRACTED ARTICLE: Linear sections of determinantal varieties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 RETRACTED ARTICLE: Linear sections of determinantal varieties.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.
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Linear sections of determinantal varieties

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Theoretical and Natural Science

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Vol.50

Article DOI

10.54254/2753-8818/50/20240645

Name of author(s) (in order)

Xiaoxi Zhou

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Xiaoxi Zhou

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Nanjing Foreign Language School

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xiaoxizhou27@gmail.com

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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.

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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)

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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.