Rocky Mountain Journal of Mathematics

Rational linear spaces on hypersurfaces over quasi-algebraically closed fields

Todd Cochrane, Craig V. Spencer, and Hee-Sung Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $k=\mathbb{F}_q(t)$ be the rational function field over $\mathbb{F}_q$ and $f(\mathbf{x}) \in k[x_1, \ldots, x_s]$ be a form of degree~$d$. For $l \in \mathbb{N}$, we establish that whenever \[ s > l + \sum_{w=1}^{d} w^2 \binom{d-w+l-1}{l-1}, \] the projective hypersurface $f(\mathbf{x})=0$ contains a $k$-rational linear space of projective dimension~$l$. We also show that if $s> 1+ d(d+1)(2d+1)/6$, then for any $k$-rational zero $\mathbf{a}$ of $f(\mathbf{x})$ there are infinitely many $s$-tuples $(\varpi_1, \ldots, \varpi_s)$ of monic irreducible polynomials over $k$, with the $\varpi_i$ not all equal, and $f(a_1\varpi_1, \ldots, a_s \varpi_s) =0$. We establish in fact more general results of the above type for systems of forms over $C_i$-fields.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1805-1816.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1422885095

Digital Object Identifier
doi:10.1216/RMJ-2014-44-6-1805

Mathematical Reviews number (MathSciNet)
MR3310949

Zentralblatt MATH identifier
1375.11033

Subjects
Primary: 11D72: Equations in many variables [See also 11P55] 12F20: Transcendental extensions 11T55: Arithmetic theory of polynomial rings over finite fields

Keywords
Diophantine equations function fields quasi-algebraic closure

Citation

Cochrane, Todd; Spencer, Craig V.; Yang, Hee-Sung. Rational linear spaces on hypersurfaces over quasi-algebraically closed fields. Rocky Mountain J. Math. 44 (2014), no. 6, 1805--1816. doi:10.1216/RMJ-2014-44-6-1805. https://projecteuclid.org/euclid.rmjm/1422885095


Export citation

References

  • M. Amer, Quadratische Formen über Funktionenkörpern, Dissertation, Mainz, 1976.
  • B.J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), 102–105.
  • J. Brüdern, R. Dietmann, J.Y. Liu and T.D. Wooley, A Birch-Goldbach theorem, Arch. Math. 94 (2010), 53–58.
  • C. Chevalley, Démonstration d'une hypothèse de M. Artin, Abh. Math. Sem. Hamburg Univ. 11 (1936), 73–75.
  • T. Cochrane, Small solutions of congruences over algebraic number fields, Illinois J. Math. 31 (1987), 618–625.
  • R. Dietmann, Linear spaces on rational hypersurfaces of odd degree, Bull. Lond. Math. Soc. 42 (2010), 891–895.
  • ––––, Systems of cubic forms, J. Lond. Math. Soc. 77 (2008), 666–686.
  • R. Dietmann and T.D. Wooley, Pairs of cubic forms in many variables, Acta Arith. 110 (2003), 125–140.
  • S. Lang, On quasi-algebraic closure, Ann. Math. 55 (1952), 373–390.
  • T.H. Lê, Green-Tao theorem in function fields, Acta Arith. 147 (2011), 129–152.
  • D.B. Leep, Systems of quadratic forms, J. reine angew. Math. 350 (1984), 109–116.
  • ––––, Amer-Brumer theorem over arbitrary fields, http://www.ms.uky.edu/$\sim$leep/Amer-Brumer_theorem.pdf (2007), 1–7.
  • D.B. Leep and W.M. Schmidt, Systems of homogeneous equations, Invent. Math. 71 (1983), 539–549.
  • D.J. Lewis and R. Schulze-Pillot, Linear spaces on the intersection of cubic hypersurfaces, Monatsh. Math. 97 (1984), 277–285.
  • M. Nagata, Note on a paper of Lang concerning quasi-algebraic closure, Mem. Coll. Sci. Univ. Kyoto 30 (1957), 237–241.
  • F.S. Roberts, Applied combinatorics, Prentice-Hall, Englewood Cliffs, NJ, 1984.
  • I.R. Shafarevich, Basic algebraic geometry 1, 2nd ed., Springer-Verlag, New York, 1994.
  • E. Warning, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley, Abh. Math. Sem. Hamburg Univ. 11 (1936), 76–83.
  • T.D. Wooley, An explicit version of Birch's theorem, Acta Arith. 85 (1998), 79–96.
  • ––––, Forms in many variables, Lond. Math. Soc. Lect. Note 247, Cambridge University Press, Cambridge, 1997.