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 ${\displaystyle x^{2}}$ in one variable is not universal, as it cannot represent negative numbers: the two-variable form ${\displaystyle x^{2}-y^{2}}$ over ${\displaystyle \mathbb {R} }$ is universal.
• Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form ${\displaystyle x^{2}+y^{2}+z^{2}+t^{2}-u^{2}}$ over ${\displaystyle \mathbb {Z} }$ 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 ${\displaystyle \mathbb {Q} }$ if and only if it is universal over Q_p for all primes ${\displaystyle p}$ (where we include ${\displaystyle p=\infty }$, letting ${\displaystyle \mathbb {Q} _{\infty }}$ denote ${\displaystyle \mathbb {R} }$).^[4] A form over ${\displaystyle \mathbb {R} }$ is universal if and only if it is not definite; a form over ${\displaystyle \mathbb {Q} _{p}}$ is universal if it has dimension at least 4.^[5] One can conclude that all indefinite forms of dimension at least 4 over ${\displaystyle \mathbb {Q} }$ are universal.^[4]

See also

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