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Universal quadratic forms over polynomial rings

김명환|Fei Xu|Yuanhua Wang
대한수학회 (The Korean Mathematical Society)|2008.09|대한수학회지 (Journal of the KMS)|제45권 제5호|pp. 1311-1322

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참고문헌

키워드

universal forms over polynomial ringsthe Four conjecture
주제분류수학

초록

The Fifteen Theorem proved by Conway and Schneeberger is a criterion for positive definite quadratic forms over the rational integer ring to be universal. In this paper, we give a proof of an analogy of the Fifteen Theorem for definite quadratic forms over polynomial rings, which is known as the Four Conjecture proposed by Gerstein.

Abstract

The Fifteen Theorem proved by Conway and Schneeberger is a criterion for positive definite quadratic forms over the rational integer ring to be universal. In this paper, we give a proof of an analogy of the Fifteen Theorem for definite quadratic forms over polynomial rings, which is known as the Four Conjecture proposed by Gerstein.

참고문헌 (12)

  1. M. Bhargava, "On the Conway-Schneeberger fifteen theorem, Quadratic forms and their applications (Dublin, 1999)", Contemporary Mathematics, 272, pp.27~37 (2000), Amer. Math. Soc.
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  3. W. Duke, "Some old problems and new results about quadratic forms", Notices of the American Mathematical Society, 44(2), pp.190~196 (1997)
  4. -lattices, Algebraic and arithmetic theory of quadratic forms, "Contemporary Mathematics", 2004, 129~134, pp.Amer. Math. Soc.
  5. M.-H. Kim, "Recent developments on universal forms, Algebraic and arithmetic theory of quadratic forms", Contemporary Mathematics, 344, pp.215~228 (2004), Amer. Math. Soc.
  6. B. M. Kim, "2-universal positive definite integral quinary quadratic forms, Integral quadratic forms and lattices (Seoul, 1998)", Contemporary Mathematics, 249, pp.51~62 (1999), Amer. Math. Soc.
  7. B. M. Kim, "A finiteness theorem for representability of quadratic forms by forms", Journal für die reine und angewandte Mathematik, 581, pp.23~30 (2005)
  8. M. Kneser, "Darstellungsmasse indefiniter quadratischer Formen", Mathematische Zeitschrift, 77, pp.188~194 (1961)
  9. W. Leahey, "Sums of squares of polynomials with coefficients in a finite field", American Mathematical Monthly, 74, pp.816~819 (1967)
  10. O. T. O'Meara, "The integral representations of quadratic forms over local fields", American Journal of Mathematics, 80, pp.843~878 (1958)
  11. O. T. O'Meara, "Introduction to Quadratic Forms" (2000), Springer-Verlag
  12. W. Schneeberger, "Arithmetic and geometry of integral lattices", 박사, pp.1995

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