Universal quadratic form

In mathematics, a universal quadratic form is a quadratic form over a ring which represents every elemnt of the ring.

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.

See also

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

References

• 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.