Universal quadratic form

In mathematics, a universal quadratic form is a quadratic form over a ring that represents every element of the ring.^[1] A non-singular form over a field which represents zero non-trivially is universal.^[2]

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 over R is universal.
• Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form x² + y² + z² + t² − u² over Z is universal.
• Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.^[3]

Forms over the rational numbers

The Hasse–Minkowski theorem implies that a form is universal over Q if and only if it is universal over Q_p 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 Q_p is universal if it has dimension at least 4.^[5] One can conclude that all indefinite forms of dimension at least 4 over Q are universal.^[4]

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. Rajwade (1993) p.146
3. Lam (2005) p.36
4. Serre (1973) p.43
5. Serre (1973) p.37

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
• Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.