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Universal quadratic form

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In mathematics, a universal quadratic form is a quadratic form over a ring which represents every element of the ring.[1]

Examples

  • Over the real numbers, the form x2 in one variable is not universal, as it cannot represent negative numbers: the two-variable form x2 - y2 is universal for R.
  • Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form x2 + y2 + z2 + t2 - u2 is universal for Z.
  • Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.[2]

See also

  • The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers.

References

  1. ^ Lam (2005) p.10
  2. ^ Lam (2005) p.36
  • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.


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