Universal quadratic form

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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]

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

See also

References

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