DOI QR코드

DOI QR Code

UNIVERSAL QUATERNARY LATTICES OVER F q[x]

  • Received : 2012.11.12
  • Accepted : 2012.11.28
  • Published : 2012.11.30

Abstract

In this paper, we show that any definite lattice over $\mathbb{F}_q[x]$ is universal if and only if it is quaternary and its discriminant is of degree 2, where $ch(\mathbb{F}_q){\neq}2$. The Four Conjecture follows as an immediate consequence.

Keywords

References

  1. Djokovic, Dragomir Z, Hermitian matrices over polynomial rings, J. Algebra 43 (1976), no. 2, 359-374. https://doi.org/10.1016/0021-8693(76)90119-8
  2. Gerstein, Larry J., On representation by quadratic ${\mathbb{F}}_q[{\chi}]-lattices$, Algebraic and arithmetic theory of quadratic forms, 129-134, Contemp. Math. 344, Amer. Math. Soc., Providence, RI, 2004.
  3. Kitaoka, Yoshiyuki Arithmetic of quadratic forms, Cambridge Tracts in Mathematics 106, Cambridge University Press, Cambridge, 1993.
  4. Kim, Myung-Hwan, Recent developments on universal forms, Algebraic and arithmetic theory of quadratic forms, 215-228, Contemp. Math. 344, Amer. Math. Soc., Providence, RI, 2004.
  5. Kim, Byeong Moon; Kim, Myung-Hwan; Oh, Byeong-Kweon, A finiteness theorem for representability of quadratic forms by forms, J. Reine Angew. Math. 581 (2005), 23-30.
  6. Kim, Myung-Hwan; Wang, Yuanhua; Xu, Fei, Universal quadratic forms over polynomial rings, J. Korean Math. Soc. 45 (2008), no. 5, 1311-1322. https://doi.org/10.4134/JKMS.2008.45.5.1311
  7. O'Meara, O. T., The integral representations of quadratic forms over local fields, Amer. J. Math. 80 (1958), 843-878. https://doi.org/10.2307/2372837
  8. O'Meara, O. T., Introduction to quadratic forms, Reprint of the 1973 edition, Classics in Mathematics. Springer-Verlag, Berlin, 2000.
  9. Chan, Wai Kiu; Daniels, Joshua, Definite regular quadratic forms over $\mathbb{F}_q$[t], Proc. Amer. Math. Soc. 133 (2005), no. 11, 3121-3131. https://doi.org/10.1090/S0002-9939-05-08197-9