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] A non-singular form over a field which represents zero non-trivially is universal.[2]

Examples[edit]

  • 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.[3]

Forms over the rational numbers[edit]

The Hasse–Minkowski theorem implies that a form is universal over Q if and only if it is universal over Qp for all p (where we include p=∞, letting Q denote R).[4] A form over R is universal if and only if it is not definite; a form over Qp is universal if it has dimension at least 4.[5] We conclude that all indefinite forms of dimension at least 4 over Q are universal.[4]

See also[edit]

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

References[edit]

  1. ^ Lam (2005) p.10
  2. ^ Rajwade (1993) p.146
  3. ^ Lam (2005) p.36
  4. ^ a b Serre (1973) p.43
  5. ^ Serre (1973) p.37